Properties

Label 3645.1.n.d.1619.1
Level $3645$
Weight $1$
Character 3645.1619
Analytic conductor $1.819$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -15
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3645,1,Mod(404,3645)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3645, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3645.404");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3645 = 3^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3645.n (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: \(1.81909197105\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.135.1
Artin image: $S_3\times C_9$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{18} + \cdots)\)

Embedding invariants

Embedding label 1619.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 3645.1619
Dual form 3645.1.n.d.2024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.939693 + 0.342020i) q^{2} +(0.173648 - 0.984808i) q^{5} +(-0.500000 - 0.866025i) q^{8} +O(q^{10})\) \(q+(0.939693 + 0.342020i) q^{2} +(0.173648 - 0.984808i) q^{5} +(-0.500000 - 0.866025i) q^{8} +(0.500000 - 0.866025i) q^{10} +(-0.173648 - 0.984808i) q^{16} +(0.500000 - 0.866025i) q^{17} +(0.500000 + 0.866025i) q^{19} +(-0.766044 - 0.642788i) q^{23} +(-0.939693 - 0.342020i) q^{25} +(-0.766044 - 0.642788i) q^{31} +(0.766044 - 0.642788i) q^{34} +(0.173648 + 0.984808i) q^{38} +(-0.939693 + 0.342020i) q^{40} +(-0.500000 - 0.866025i) q^{46} +(1.53209 - 1.28558i) q^{47} +(0.173648 - 0.984808i) q^{49} +(-0.766044 - 0.642788i) q^{50} -1.00000 q^{53} +(-0.766044 + 0.642788i) q^{61} +(-0.500000 - 0.866025i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(0.939693 + 0.342020i) q^{79} -1.00000 q^{80} +(0.939693 + 0.342020i) q^{83} +(-0.766044 - 0.642788i) q^{85} +(1.87939 - 0.684040i) q^{94} +(0.939693 - 0.342020i) q^{95} +(0.500000 - 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{8} + 3 q^{10} + 3 q^{17} + 3 q^{19} - 3 q^{46} - 6 q^{53} - 3 q^{62} - 3 q^{64} - 6 q^{80} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3645\mathbb{Z}\right)^\times\).

\(n\) \(731\) \(2917\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 0.173648 0.984808i 0.173648 0.984808i
\(6\) 0 0
\(7\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(8\) −0.500000 0.866025i −0.500000 0.866025i
\(9\) 0 0
\(10\) 0.500000 0.866025i 0.500000 0.866025i
\(11\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(12\) 0 0
\(13\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.173648 0.984808i −0.173648 0.984808i
\(17\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(18\) 0 0
\(19\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(24\) 0 0
\(25\) −0.939693 0.342020i −0.939693 0.342020i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(30\) 0 0
\(31\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0.766044 0.642788i 0.766044 0.642788i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(39\) 0 0
\(40\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(41\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(42\) 0 0
\(43\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.500000 0.866025i −0.500000 0.866025i
\(47\) 1.53209 1.28558i 1.53209 1.28558i 0.766044 0.642788i \(-0.222222\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(48\) 0 0
\(49\) 0.173648 0.984808i 0.173648 0.984808i
\(50\) −0.766044 0.642788i −0.766044 0.642788i
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(60\) 0 0
\(61\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(62\) −0.500000 0.866025i −0.500000 0.866025i
\(63\) 0 0
\(64\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(80\) −1.00000 −1.00000
\(81\) 0 0
\(82\) 0 0
\(83\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(84\) 0 0
\(85\) −0.766044 0.642788i −0.766044 0.642788i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 1.87939 0.684040i 1.87939 0.684040i
\(95\) 0.939693 0.342020i 0.939693 0.342020i
\(96\) 0 0
\(97\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(98\) 0.500000 0.866025i 0.500000 0.866025i
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(102\) 0 0
\(103\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −0.939693 0.342020i −0.939693 0.342020i
\(107\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(108\) 0 0
\(109\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.347296 1.96962i 0.347296 1.96962i 0.173648 0.984808i \(-0.444444\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(114\) 0 0
\(115\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(122\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(126\) 0 0
\(127\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(128\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −1.00000 −1.00000
\(137\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(138\) 0 0
\(139\) 1.53209 + 1.28558i 1.53209 + 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(150\) 0 0
\(151\) 0.347296 + 1.96962i 0.347296 + 1.96962i 0.173648 + 0.984808i \(0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(152\) 0.500000 0.866025i 0.500000 0.866025i
\(153\) 0 0
\(154\) 0 0
\(155\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(156\) 0 0
\(157\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(158\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(167\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(168\) 0 0
\(169\) 0.766044 0.642788i 0.766044 0.642788i
\(170\) −0.500000 0.866025i −0.500000 0.866025i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 1.00000 1.00000
\(191\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(192\) 0 0
\(193\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(198\) 0 0
\(199\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.87939 + 0.684040i 1.87939 + 0.684040i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.939693 0.342020i −0.939693 0.342020i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.00000 1.73205i 1.00000 1.73205i
\(227\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(228\) 0 0
\(229\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(230\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(231\) 0 0
\(232\) 0 0
\(233\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) 0 0
\(235\) −1.00000 1.73205i −1.00000 1.73205i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(240\) 0 0
\(241\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(242\) −1.00000 −1.00000
\(243\) 0 0
\(244\) 0 0
\(245\) −0.939693 0.342020i −0.939693 0.342020i
\(246\) 0 0
\(247\) 0 0
\(248\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(249\) 0 0
\(250\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(251\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.53209 1.28558i 1.53209 1.28558i 0.766044 0.642788i \(-0.222222\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(264\) 0 0
\(265\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) −0.939693 0.342020i −0.939693 0.342020i
\(273\) 0 0
\(274\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(278\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(282\) 0 0
\(283\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −0.347296 + 1.96962i −0.347296 + 1.96962i
\(303\) 0 0
\(304\) 0.766044 0.642788i 0.766044 0.642788i
\(305\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(306\) 0 0
\(307\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(311\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(312\) 0 0
\(313\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(321\) 0 0
\(322\) 0 0
\(323\) 1.00000 1.00000
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.53209 1.28558i 1.53209 1.28558i 0.766044 0.642788i \(-0.222222\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(338\) 0.939693 0.342020i 0.939693 0.342020i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0.173648 0.984808i 0.173648 0.984808i
\(347\) 1.53209 + 1.28558i 1.53209 + 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(348\) 0 0
\(349\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.87939 0.684040i −1.87939 0.684040i −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 0.342020i \(-0.888889\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(368\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.87939 0.684040i −1.87939 0.684040i
\(377\) 0 0
\(378\) 0 0
\(379\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(390\) 0 0
\(391\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(392\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(393\) 0 0
\(394\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(395\) 0.500000 0.866025i 0.500000 0.866025i
\(396\) 0 0
\(397\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(398\) −1.53209 + 1.28558i −1.53209 + 1.28558i
\(399\) 0 0
\(400\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(401\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.500000 0.866025i 0.500000 0.866025i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(420\) 0 0
\(421\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(422\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(423\) 0 0
\(424\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(425\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.173648 0.984808i 0.173648 0.984808i
\(438\) 0 0
\(439\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0.173648 0.984808i 0.173648 0.984808i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(458\) 1.00000 1.00000
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(462\) 0 0
\(463\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −1.53209 + 1.28558i −1.53209 + 1.28558i
\(467\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −0.347296 1.96962i −0.347296 1.96962i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.173648 0.984808i −0.173648 0.984808i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(489\) 0 0
\(490\) −0.766044 0.642788i −0.766044 0.642788i
\(491\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 1.00000
\(513\) 0 0
\(514\) 1.00000 1.00000
\(515\) 0 0
\(516\) 0 0
\(517\) 0 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 \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 1.87939 0.684040i 1.87939 0.684040i
\(527\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(528\) 0 0
\(529\) 0 0
\(530\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0.347296 1.96962i 0.347296 1.96962i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(542\) −0.939693 0.342020i −0.939693 0.342020i
\(543\) 0 0
\(544\) 0 0
\(545\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(546\) 0 0
\(547\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.53209 + 1.28558i 1.53209 + 1.28558i 0.766044 + 0.642788i \(0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(564\) 0 0
\(565\) −1.87939 0.684040i −1.87939 0.684040i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(570\) 0 0
\(571\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(576\) 0 0
\(577\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −0.500000 0.866025i −0.500000 0.866025i
\(587\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(588\) 0 0
\(589\) 0.173648 0.984808i 0.173648 0.984808i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(600\) 0 0
\(601\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(606\) 0 0
\(607\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(618\) 0 0
\(619\) −1.87939 0.684040i −1.87939 0.684040i −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 0.342020i \(-0.888889\pi\)
\(620\) 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 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(632\) −0.173648 0.984808i −0.173648 0.984808i
\(633\) 0 0
\(634\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(641\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(642\) 0 0
\(643\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(647\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(660\) 0 0
\(661\) −1.87939 + 0.684040i −1.87939 + 0.684040i −0.939693 + 0.342020i \(0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(662\) 1.87939 0.684040i 1.87939 0.684040i
\(663\) 0 0
\(664\) −0.173648 0.984808i −0.173648 0.984808i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.87939 0.684040i −1.87939 0.684040i −0.939693 0.342020i \(-0.888889\pi\)
−0.939693 0.342020i \(-0.888889\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(681\) 0 0
\(682\) 0 0
\(683\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) 0.500000 0.866025i 0.500000 0.866025i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(695\) 1.53209 1.28558i 1.53209 1.28558i
\(696\) 0 0
\(697\) 0 0
\(698\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −1.53209 1.28558i −1.53209 1.28558i
\(707\) 0 0
\(708\) 0 0
\(709\) 1.53209 1.28558i 1.53209 1.28558i 0.766044 0.642788i \(-0.222222\pi\)
0.766044 0.642788i \(-0.222222\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.87939 + 0.684040i −1.87939 + 0.684040i −0.939693 + 0.342020i \(0.888889\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(752\) −1.53209 1.28558i −1.53209 1.28558i
\(753\) 0 0
\(754\) 0 0
\(755\) 2.00000 2.00000
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −0.939693 0.342020i −0.939693 0.342020i
\(759\) 0 0
\(760\) −0.766044 0.642788i −0.766044 0.642788i
\(761\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(767\) 0 0
\(768\) 0 0
\(769\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(774\) 0 0
\(775\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −1.00000 −1.00000
\(783\) 0 0
\(784\) −1.00000 −1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0.766044 0.642788i 0.766044 0.642788i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.939693 0.342020i 0.939693 0.342020i 0.173648 0.984808i \(-0.444444\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(798\) 0 0
\(799\) −0.347296 1.96962i −0.347296 1.96962i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −0.500000 0.866025i −0.500000 0.866025i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(822\) 0 0
\(823\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(828\) 0 0
\(829\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(830\) 0.766044 0.642788i 0.766044 0.642788i
\(831\) 0 0
\(832\) 0 0
\(833\) −0.766044 0.642788i −0.766044 0.642788i
\(834\) 0 0
\(835\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(840\) 0 0
\(841\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(842\) 0.173648 0.984808i 0.173648 0.984808i
\(843\) 0 0
\(844\) 0 0
\(845\) −0.500000 0.866025i −0.500000 0.866025i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(849\) 0 0
\(850\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.00000 1.73205i −1.00000 1.73205i
\(857\) −0.766044 + 0.642788i −0.766044 + 0.642788i −0.939693 0.342020i \(-0.888889\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(858\) 0 0
\(859\) −0.173648 + 0.984808i −0.173648 + 0.984808i 0.766044 + 0.642788i \(0.222222\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(864\) 0 0
\(865\) −1.00000 −1.00000
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(873\) 0 0
\(874\) 0.500000 0.866025i 0.500000 0.866025i
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(878\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) 0 0
\(883\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.173648 0.984808i 0.173648 0.984808i
\(887\) −0.766044 0.642788i −0.766044 0.642788i 0.173648 0.984808i \(-0.444444\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.87939 + 0.684040i 1.87939 + 0.684040i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(902\) 0 0
\(903\) 0 0
\(904\) −1.87939 + 0.684040i −1.87939 + 0.684040i
\(905\) 0.939693 0.342020i 0.939693 0.342020i
\(906\) 0 0
\(907\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(920\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(921\) 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 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(930\) 0 0
\(931\) 0.939693 0.342020i 0.939693 0.342020i
\(932\) 0 0
\(933\) 0 0
\(934\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.939693 + 0.342020i 0.939693 + 0.342020i 0.766044 0.642788i \(-0.222222\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.173648 0.984808i 0.173648 0.984808i
\(951\) 0 0
\(952\) 0 0
\(953\) −1.00000 1.73205i −1.00000 1.73205i −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
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(968\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(977\) 0.347296 1.96962i 0.347296 1.96962i 0.173648 0.984808i \(-0.444444\pi\)
0.173648 0.984808i \(-0.444444\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −0.173648 0.984808i −0.173648 0.984808i −0.939693 0.342020i \(-0.888889\pi\)
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 0
\(990\) 0 0
\(991\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.53209 + 1.28558i 1.53209 + 1.28558i
\(996\) 0 0
\(997\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(998\) 1.00000 1.00000
\(999\) 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 3645.1.n.d.1619.1 6
3.2 odd 2 3645.1.n.e.1619.1 6
5.4 even 2 3645.1.n.e.1619.1 6
9.2 odd 6 3645.1.n.e.2834.1 6
9.4 even 3 inner 3645.1.n.d.404.1 6
9.5 odd 6 3645.1.n.e.404.1 6
9.7 even 3 inner 3645.1.n.d.2834.1 6
15.14 odd 2 CM 3645.1.n.d.1619.1 6
27.2 odd 18 3645.1.n.e.3239.1 6
27.4 even 9 405.1.h.b.134.1 2
27.5 odd 18 405.1.h.a.269.1 2
27.7 even 9 inner 3645.1.n.d.2024.1 6
27.11 odd 18 3645.1.n.e.809.1 6
27.13 even 9 135.1.d.a.134.1 1
27.14 odd 18 135.1.d.b.134.1 yes 1
27.16 even 9 inner 3645.1.n.d.809.1 6
27.20 odd 18 3645.1.n.e.2024.1 6
27.22 even 9 405.1.h.b.269.1 2
27.23 odd 18 405.1.h.a.134.1 2
27.25 even 9 inner 3645.1.n.d.3239.1 6
45.4 even 6 3645.1.n.e.404.1 6
45.14 odd 6 inner 3645.1.n.d.404.1 6
45.29 odd 6 inner 3645.1.n.d.2834.1 6
45.34 even 6 3645.1.n.e.2834.1 6
108.67 odd 18 2160.1.c.b.1889.1 1
108.95 even 18 2160.1.c.a.1889.1 1
135.4 even 18 405.1.h.a.134.1 2
135.13 odd 36 675.1.c.c.26.2 2
135.14 odd 18 135.1.d.a.134.1 1
135.22 odd 36 2025.1.j.c.26.2 4
135.23 even 36 2025.1.j.c.701.1 4
135.29 odd 18 inner 3645.1.n.d.3239.1 6
135.32 even 36 2025.1.j.c.26.1 4
135.34 even 18 3645.1.n.e.2024.1 6
135.49 even 18 405.1.h.a.269.1 2
135.58 odd 36 2025.1.j.c.701.2 4
135.59 odd 18 405.1.h.b.269.1 2
135.67 odd 36 675.1.c.c.26.1 2
135.68 even 36 675.1.c.c.26.1 2
135.74 odd 18 inner 3645.1.n.d.2024.1 6
135.77 even 36 2025.1.j.c.701.2 4
135.79 even 18 3645.1.n.e.3239.1 6
135.94 even 18 135.1.d.b.134.1 yes 1
135.103 odd 36 2025.1.j.c.26.1 4
135.104 odd 18 405.1.h.b.134.1 2
135.112 odd 36 2025.1.j.c.701.1 4
135.113 even 36 2025.1.j.c.26.2 4
135.119 odd 18 inner 3645.1.n.d.809.1 6
135.122 even 36 675.1.c.c.26.2 2
135.124 even 18 3645.1.n.e.809.1 6
540.419 even 18 2160.1.c.b.1889.1 1
540.499 odd 18 2160.1.c.a.1889.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.1.d.a.134.1 1 27.13 even 9
135.1.d.a.134.1 1 135.14 odd 18
135.1.d.b.134.1 yes 1 27.14 odd 18
135.1.d.b.134.1 yes 1 135.94 even 18
405.1.h.a.134.1 2 27.23 odd 18
405.1.h.a.134.1 2 135.4 even 18
405.1.h.a.269.1 2 27.5 odd 18
405.1.h.a.269.1 2 135.49 even 18
405.1.h.b.134.1 2 27.4 even 9
405.1.h.b.134.1 2 135.104 odd 18
405.1.h.b.269.1 2 27.22 even 9
405.1.h.b.269.1 2 135.59 odd 18
675.1.c.c.26.1 2 135.67 odd 36
675.1.c.c.26.1 2 135.68 even 36
675.1.c.c.26.2 2 135.13 odd 36
675.1.c.c.26.2 2 135.122 even 36
2025.1.j.c.26.1 4 135.32 even 36
2025.1.j.c.26.1 4 135.103 odd 36
2025.1.j.c.26.2 4 135.22 odd 36
2025.1.j.c.26.2 4 135.113 even 36
2025.1.j.c.701.1 4 135.23 even 36
2025.1.j.c.701.1 4 135.112 odd 36
2025.1.j.c.701.2 4 135.58 odd 36
2025.1.j.c.701.2 4 135.77 even 36
2160.1.c.a.1889.1 1 108.95 even 18
2160.1.c.a.1889.1 1 540.499 odd 18
2160.1.c.b.1889.1 1 108.67 odd 18
2160.1.c.b.1889.1 1 540.419 even 18
3645.1.n.d.404.1 6 9.4 even 3 inner
3645.1.n.d.404.1 6 45.14 odd 6 inner
3645.1.n.d.809.1 6 27.16 even 9 inner
3645.1.n.d.809.1 6 135.119 odd 18 inner
3645.1.n.d.1619.1 6 1.1 even 1 trivial
3645.1.n.d.1619.1 6 15.14 odd 2 CM
3645.1.n.d.2024.1 6 27.7 even 9 inner
3645.1.n.d.2024.1 6 135.74 odd 18 inner
3645.1.n.d.2834.1 6 9.7 even 3 inner
3645.1.n.d.2834.1 6 45.29 odd 6 inner
3645.1.n.d.3239.1 6 27.25 even 9 inner
3645.1.n.d.3239.1 6 135.29 odd 18 inner
3645.1.n.e.404.1 6 9.5 odd 6
3645.1.n.e.404.1 6 45.4 even 6
3645.1.n.e.809.1 6 27.11 odd 18
3645.1.n.e.809.1 6 135.124 even 18
3645.1.n.e.1619.1 6 3.2 odd 2
3645.1.n.e.1619.1 6 5.4 even 2
3645.1.n.e.2024.1 6 27.20 odd 18
3645.1.n.e.2024.1 6 135.34 even 18
3645.1.n.e.2834.1 6 9.2 odd 6
3645.1.n.e.2834.1 6 45.34 even 6
3645.1.n.e.3239.1 6 27.2 odd 18
3645.1.n.e.3239.1 6 135.79 even 18