Properties

Label 2527.1.be.a.1199.1
Level $2527$
Weight $1$
Character 2527.1199
Analytic conductor $1.261$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -19
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2527,1,Mod(333,2527)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2527, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([12, 17]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2527.333");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2527 = 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2527.be (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.26113728692\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.931.1
Artin image: $C_3^2:C_{18}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{54} - \cdots)\)

Embedding invariants

Embedding label 1199.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 2527.1199
Dual form 2527.1.be.a.333.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.173648 + 0.984808i) q^{4} +(-0.173648 + 0.984808i) q^{5} +1.00000 q^{7} +(-0.939693 - 0.342020i) q^{9} +O(q^{10})\) \(q+(0.173648 + 0.984808i) q^{4} +(-0.173648 + 0.984808i) q^{5} +1.00000 q^{7} +(-0.939693 - 0.342020i) q^{9} -1.00000 q^{11} +(-0.939693 + 0.342020i) q^{16} +(-1.87939 + 0.684040i) q^{17} -1.00000 q^{20} +(-0.766044 + 0.642788i) q^{23} +(0.173648 + 0.984808i) q^{28} +(-0.173648 + 0.984808i) q^{35} +(0.173648 - 0.984808i) q^{36} +(0.939693 - 0.342020i) q^{43} +(-0.173648 - 0.984808i) q^{44} +(0.500000 - 0.866025i) q^{45} +(0.939693 + 0.342020i) q^{47} +1.00000 q^{49} +(0.173648 - 0.984808i) q^{55} +(-0.766044 + 0.642788i) q^{61} +(-0.939693 - 0.342020i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(-1.00000 - 1.73205i) q^{68} +(-0.173648 + 0.984808i) q^{73} -1.00000 q^{77} +(-0.173648 - 0.984808i) q^{80} +(0.766044 + 0.642788i) q^{81} +(0.500000 + 0.866025i) q^{83} +(-0.347296 - 1.96962i) q^{85} +(-0.766044 - 0.642788i) q^{92} +(0.939693 + 0.342020i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{7} - 6 q^{11} - 6 q^{20} + 3 q^{45} + 6 q^{49} - 3 q^{64} - 6 q^{68} - 6 q^{77} + 3 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2527\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(1807\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{18}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(3\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(4\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(5\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(6\) 0 0
\(7\) 1.00000 1.00000
\(8\) 0 0
\(9\) −0.939693 0.342020i −0.939693 0.342020i
\(10\) 0 0
\(11\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 0 0
\(13\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(17\) −1.87939 + 0.684040i −1.87939 + 0.684040i −0.939693 + 0.342020i \(0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) −1.00000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(29\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(36\) 0.173648 0.984808i 0.173648 0.984808i
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(42\) 0 0
\(43\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(44\) −0.173648 0.984808i −0.173648 0.984808i
\(45\) 0.500000 0.866025i 0.500000 0.866025i
\(46\) 0 0
\(47\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(54\) 0 0
\(55\) 0.173648 0.984808i 0.173648 0.984808i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(60\) 0 0
\(61\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(62\) 0 0
\(63\) −0.939693 0.342020i −0.939693 0.342020i
\(64\) −0.500000 0.866025i −0.500000 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(68\) −1.00000 1.73205i −1.00000 1.73205i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(72\) 0 0
\(73\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 −1.00000
\(78\) 0 0
\(79\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(80\) −0.173648 0.984808i −0.173648 0.984808i
\(81\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(82\) 0 0
\(83\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(84\) 0 0
\(85\) −0.347296 1.96962i −0.347296 1.96962i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.766044 0.642788i −0.766044 0.642788i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(98\) 0 0
\(99\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(100\) 0 0
\(101\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(102\) 0 0
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −0.500000 0.866025i −0.500000 0.866025i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.87939 + 0.684040i −1.87939 + 0.684040i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(126\) 0 0
\(127\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.53209 + 1.28558i 1.53209 + 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(138\) 0 0
\(139\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(140\) −1.00000 −1.00000
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(150\) 0 0
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 2.00000 2.00000
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(162\) 0 0
\(163\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(168\) 0 0
\(169\) 0.173648 0.984808i 0.173648 0.984808i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(173\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.939693 0.342020i 0.939693 0.342020i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(181\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.87939 0.684040i 1.87939 0.684040i
\(188\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(189\) 0 0
\(190\) 0 0
\(191\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(192\) 0 0
\(193\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(197\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.939693 0.342020i 0.939693 0.342020i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 1.00000 1.00000
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) −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 0
\(232\) 0 0
\(233\) 0.347296 1.96962i 0.347296 1.96962i 0.173648 0.984808i \(-0.444444\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(234\) 0 0
\(235\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −0.766044 0.642788i −0.766044 0.642788i
\(245\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(252\) 0.173648 0.984808i 0.173648 0.984808i
\(253\) 0.766044 0.642788i 0.766044 0.642788i
\(254\) 0 0
\(255\) 0 0
\(256\) 0.766044 0.642788i 0.766044 0.642788i
\(257\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.347296 1.96962i 0.347296 1.96962i 0.173648 0.984808i \(-0.444444\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(270\) 0 0
\(271\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(272\) 1.53209 1.28558i 1.53209 1.28558i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(282\) 0 0
\(283\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.29813 1.92836i 2.29813 1.92836i
\(290\) 0 0
\(291\) 0 0
\(292\) −1.00000 −1.00000
\(293\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.939693 0.342020i 0.939693 0.342020i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.500000 0.866025i −0.500000 0.866025i
\(306\) 0 0
\(307\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(308\) −0.173648 0.984808i −0.173648 0.984808i
\(309\) 0 0
\(310\) 0 0
\(311\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(314\) 0 0
\(315\) 0.500000 0.866025i 0.500000 0.866025i
\(316\) 0 0
\(317\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.939693 0.342020i 0.939693 0.342020i
\(321\) 0 0
\(322\) 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
\(329\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(330\) 0 0
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 1.87939 0.684040i 1.87939 0.684040i
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(348\) 0 0
\(349\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.939693 0.342020i −0.939693 0.342020i
\(366\) 0 0
\(367\) −1.87939 0.684040i −1.87939 0.684040i −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 0.342020i \(-0.888889\pi\)
\(368\) 0.500000 0.866025i 0.500000 0.866025i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(384\) 0 0
\(385\) 0.173648 0.984808i 0.173648 0.984808i
\(386\) 0 0
\(387\) −1.00000 −1.00000
\(388\) 0 0
\(389\) 1.53209 1.28558i 1.53209 1.28558i 0.766044 0.642788i \(-0.222222\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(390\) 0 0
\(391\) 1.00000 1.73205i 1.00000 1.73205i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(397\) 1.53209 + 1.28558i 1.53209 + 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(405\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(422\) 0 0
\(423\) −0.766044 0.642788i −0.766044 0.642788i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(432\) 0 0
\(433\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(440\) 0 0
\(441\) −0.939693 0.342020i −0.939693 0.342020i
\(442\) 0 0
\(443\) 0.347296 + 1.96962i 0.347296 + 1.96962i 0.173648 + 0.984808i \(0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.500000 0.866025i −0.500000 0.866025i
\(449\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0.766044 0.642788i 0.766044 0.642788i
\(461\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(462\) 0 0
\(463\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(474\) 0 0
\(475\) 0 0
\(476\) −1.00000 1.73205i −1.00000 1.73205i
\(477\) 0 0
\(478\) 0 0
\(479\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(500\) −0.939693 0.342020i −0.939693 0.342020i
\(501\) 0 0
\(502\) 0 0
\(503\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(504\) 0 0
\(505\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(510\) 0 0
\(511\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.939693 0.342020i −0.939693 0.342020i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(524\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.00000 −1.00000
\(540\) 0 0
\(541\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(548\) 0.939693 0.342020i 0.939693 0.342020i
\(549\) 0.939693 0.342020i 0.939693 0.342020i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.766044 0.642788i −0.766044 0.642788i
\(557\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.173648 0.984808i −0.173648 0.984808i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(577\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.347296 1.96962i 0.347296 1.96962i 0.173648 0.984808i \(-0.444444\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(594\) 0 0
\(595\) −0.347296 1.96962i −0.347296 1.96962i
\(596\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(600\) 0 0
\(601\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 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
\(611\) 0 0
\(612\) 0.347296 + 1.96962i 0.347296 + 1.96962i
\(613\) 1.53209 + 1.28558i 1.53209 + 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(618\) 0 0
\(619\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.766044 0.642788i −0.766044 0.642788i
\(626\) 0 0
\(627\) 0 0
\(628\) 0.500000 0.866025i 0.500000 0.866025i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(642\) 0 0
\(643\) 1.53209 + 1.28558i 1.53209 + 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(644\) −0.766044 0.642788i −0.766044 0.642788i
\(645\) 0 0
\(646\) 0 0
\(647\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(653\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(654\) 0 0
\(655\) −1.53209 + 1.28558i −1.53209 + 1.28558i
\(656\) 0 0
\(657\) 0.500000 0.866025i 0.500000 0.866025i
\(658\) 0 0
\(659\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(660\) 0 0
\(661\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.766044 0.642788i 0.766044 0.642788i
\(672\) 0 0
\(673\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 0 0
\(685\) 1.00000 1.00000
\(686\) 0 0
\(687\) 0 0
\(688\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(689\) 0 0
\(690\) 0 0
\(691\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(692\) 0 0
\(693\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(694\) 0 0
\(695\) −0.500000 0.866025i −0.500000 0.866025i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(705\) 0 0
\(706\) 0 0
\(707\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(708\) 0 0
\(709\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.87939 + 0.684040i −1.87939 + 0.684040i −0.939693 + 0.342020i \(0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(720\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(728\) 0 0
\(729\) −0.500000 0.866025i −0.500000 0.866025i
\(730\) 0 0
\(731\) −1.53209 + 1.28558i −1.53209 + 1.28558i
\(732\) 0 0
\(733\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.87939 + 0.684040i −1.87939 + 0.684040i −0.939693 + 0.342020i \(0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(744\) 0 0
\(745\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(746\) 0 0
\(747\) −0.173648 0.984808i −0.173648 0.984808i
\(748\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(752\) −1.00000 −1.00000
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 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
\(764\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(765\) −0.347296 + 1.96962i −0.347296 + 1.96962i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(785\) 0.766044 0.642788i 0.766044 0.642788i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −0.766044 0.642788i −0.766044 0.642788i
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) −2.00000 −2.00000
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0.173648 0.984808i 0.173648 0.984808i
\(804\) 0 0
\(805\) −0.500000 0.866025i −0.500000 0.866025i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(822\) 0 0
\(823\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(828\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(829\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.87939 + 0.684040i −1.87939 + 0.684040i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(840\) 0 0
\(841\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(858\) 0 0
\(859\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(860\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(876\) 0 0
\(877\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(881\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(882\) 0 0
\(883\) 0.347296 + 1.96962i 0.347296 + 1.96962i 0.173648 + 0.984808i \(0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.766044 0.642788i −0.766044 0.642788i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(908\) 0 0
\(909\) −0.766044 0.642788i −0.766044 0.642788i
\(910\) 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 0
\(915\) 0 0
\(916\) −1.87939 0.684040i −1.87939 0.684040i
\(917\) 1.53209 + 1.28558i 1.53209 + 1.28558i
\(918\) 0 0
\(919\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.00000 2.00000
\(933\) 0 0
\(934\) 0 0
\(935\) 0.347296 + 1.96962i 0.347296 + 1.96962i
\(936\) 0 0
\(937\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.939693 0.342020i −0.939693 0.342020i
\(941\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.87939 + 0.684040i −1.87939 + 0.684040i −0.939693 + 0.342020i \(0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(954\) 0 0
\(955\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(956\) −1.87939 0.684040i −1.87939 0.684040i
\(957\) 0 0
\(958\) 0 0
\(959\) −0.173648 0.984808i −0.173648 0.984808i
\(960\) 0 0
\(961\) −0.500000 0.866025i −0.500000 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0.347296 1.96962i 0.347296 1.96962i 0.173648 0.984808i \(-0.444444\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(972\) 0 0
\(973\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(974\) 0 0
\(975\) 0 0
\(976\) 0.500000 0.866025i 0.500000 0.866025i
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.00000 −1.00000
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(984\) 0 0
\(985\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(990\) 0 0
\(991\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.500000 0.866025i −0.500000 0.866025i
\(996\) 0 0
\(997\) 1.53209 1.28558i 1.53209 1.28558i 0.766044 0.642788i \(-0.222222\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(998\) 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 2527.1.be.a.1199.1 6
7.4 even 3 2527.1.bd.a.116.1 6
19.2 odd 18 2527.1.n.a.2459.1 2
19.3 odd 18 133.1.r.a.37.1 yes 2
19.4 even 9 2527.1.bd.a.849.1 6
19.5 even 9 2527.1.j.a.2235.1 2
19.6 even 9 2527.1.bd.a.2067.1 6
19.7 even 3 inner 2527.1.be.a.2473.1 6
19.8 odd 6 inner 2527.1.be.a.660.1 6
19.9 even 9 2527.1.bd.a.1416.1 6
19.10 odd 18 2527.1.bd.a.1416.1 6
19.11 even 3 inner 2527.1.be.a.660.1 6
19.12 odd 6 inner 2527.1.be.a.2473.1 6
19.13 odd 18 2527.1.bd.a.2067.1 6
19.14 odd 18 2527.1.j.a.2235.1 2
19.15 odd 18 2527.1.bd.a.849.1 6
19.16 even 9 133.1.r.a.37.1 yes 2
19.17 even 9 2527.1.n.a.2459.1 2
19.18 odd 2 CM 2527.1.be.a.1199.1 6
57.35 odd 18 1197.1.cz.a.37.1 2
57.41 even 18 1197.1.cz.a.37.1 2
76.3 even 18 2128.1.cl.c.1633.1 2
76.35 odd 18 2128.1.cl.c.1633.1 2
95.3 even 36 3325.1.y.a.1899.2 4
95.22 even 36 3325.1.y.a.1899.1 4
95.54 even 18 3325.1.bm.a.1101.1 2
95.73 odd 36 3325.1.y.a.1899.2 4
95.79 odd 18 3325.1.bm.a.1101.1 2
95.92 odd 36 3325.1.y.a.1899.1 4
133.3 even 18 931.1.r.a.18.1 2
133.4 even 9 inner 2527.1.be.a.2293.1 6
133.11 even 3 2527.1.bd.a.2104.1 6
133.16 even 9 931.1.b.a.246.1 1
133.18 odd 6 2527.1.bd.a.116.1 6
133.25 even 9 inner 2527.1.be.a.984.1 6
133.32 odd 18 inner 2527.1.be.a.984.1 6
133.41 even 18 931.1.r.a.569.1 2
133.46 odd 6 2527.1.bd.a.2104.1 6
133.53 odd 18 inner 2527.1.be.a.2293.1 6
133.54 odd 18 931.1.b.b.246.1 1
133.60 odd 18 133.1.r.a.18.1 2
133.67 odd 18 inner 2527.1.be.a.333.1 6
133.73 odd 18 931.1.r.a.18.1 2
133.74 even 9 2527.1.j.a.1376.1 2
133.79 odd 18 931.1.b.a.246.1 1
133.81 even 9 2527.1.n.a.1152.1 2
133.88 odd 6 2527.1.bd.a.1390.1 6
133.102 even 3 2527.1.bd.a.1390.1 6
133.109 odd 18 2527.1.n.a.1152.1 2
133.111 odd 18 931.1.r.a.569.1 2
133.116 odd 18 2527.1.j.a.1376.1 2
133.117 even 18 931.1.b.b.246.1 1
133.123 even 9 inner 2527.1.be.a.333.1 6
133.130 even 9 133.1.r.a.18.1 2
399.263 odd 18 1197.1.cz.a.550.1 2
399.326 even 18 1197.1.cz.a.550.1 2
532.263 odd 18 2128.1.cl.c.417.1 2
532.459 even 18 2128.1.cl.c.417.1 2
665.193 even 36 3325.1.y.a.949.1 4
665.263 odd 36 3325.1.y.a.949.1 4
665.459 odd 18 3325.1.bm.a.151.1 2
665.529 even 18 3325.1.bm.a.151.1 2
665.592 even 36 3325.1.y.a.949.2 4
665.662 odd 36 3325.1.y.a.949.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
133.1.r.a.18.1 2 133.60 odd 18
133.1.r.a.18.1 2 133.130 even 9
133.1.r.a.37.1 yes 2 19.3 odd 18
133.1.r.a.37.1 yes 2 19.16 even 9
931.1.b.a.246.1 1 133.16 even 9
931.1.b.a.246.1 1 133.79 odd 18
931.1.b.b.246.1 1 133.54 odd 18
931.1.b.b.246.1 1 133.117 even 18
931.1.r.a.18.1 2 133.3 even 18
931.1.r.a.18.1 2 133.73 odd 18
931.1.r.a.569.1 2 133.41 even 18
931.1.r.a.569.1 2 133.111 odd 18
1197.1.cz.a.37.1 2 57.35 odd 18
1197.1.cz.a.37.1 2 57.41 even 18
1197.1.cz.a.550.1 2 399.263 odd 18
1197.1.cz.a.550.1 2 399.326 even 18
2128.1.cl.c.417.1 2 532.263 odd 18
2128.1.cl.c.417.1 2 532.459 even 18
2128.1.cl.c.1633.1 2 76.3 even 18
2128.1.cl.c.1633.1 2 76.35 odd 18
2527.1.j.a.1376.1 2 133.74 even 9
2527.1.j.a.1376.1 2 133.116 odd 18
2527.1.j.a.2235.1 2 19.5 even 9
2527.1.j.a.2235.1 2 19.14 odd 18
2527.1.n.a.1152.1 2 133.81 even 9
2527.1.n.a.1152.1 2 133.109 odd 18
2527.1.n.a.2459.1 2 19.2 odd 18
2527.1.n.a.2459.1 2 19.17 even 9
2527.1.bd.a.116.1 6 7.4 even 3
2527.1.bd.a.116.1 6 133.18 odd 6
2527.1.bd.a.849.1 6 19.4 even 9
2527.1.bd.a.849.1 6 19.15 odd 18
2527.1.bd.a.1390.1 6 133.88 odd 6
2527.1.bd.a.1390.1 6 133.102 even 3
2527.1.bd.a.1416.1 6 19.9 even 9
2527.1.bd.a.1416.1 6 19.10 odd 18
2527.1.bd.a.2067.1 6 19.6 even 9
2527.1.bd.a.2067.1 6 19.13 odd 18
2527.1.bd.a.2104.1 6 133.11 even 3
2527.1.bd.a.2104.1 6 133.46 odd 6
2527.1.be.a.333.1 6 133.67 odd 18 inner
2527.1.be.a.333.1 6 133.123 even 9 inner
2527.1.be.a.660.1 6 19.8 odd 6 inner
2527.1.be.a.660.1 6 19.11 even 3 inner
2527.1.be.a.984.1 6 133.25 even 9 inner
2527.1.be.a.984.1 6 133.32 odd 18 inner
2527.1.be.a.1199.1 6 1.1 even 1 trivial
2527.1.be.a.1199.1 6 19.18 odd 2 CM
2527.1.be.a.2293.1 6 133.4 even 9 inner
2527.1.be.a.2293.1 6 133.53 odd 18 inner
2527.1.be.a.2473.1 6 19.7 even 3 inner
2527.1.be.a.2473.1 6 19.12 odd 6 inner
3325.1.y.a.949.1 4 665.193 even 36
3325.1.y.a.949.1 4 665.263 odd 36
3325.1.y.a.949.2 4 665.592 even 36
3325.1.y.a.949.2 4 665.662 odd 36
3325.1.y.a.1899.1 4 95.22 even 36
3325.1.y.a.1899.1 4 95.92 odd 36
3325.1.y.a.1899.2 4 95.3 even 36
3325.1.y.a.1899.2 4 95.73 odd 36
3325.1.bm.a.151.1 2 665.459 odd 18
3325.1.bm.a.151.1 2 665.529 even 18
3325.1.bm.a.1101.1 2 95.54 even 18
3325.1.bm.a.1101.1 2 95.79 odd 18