Properties

Label 1444.1.g.a.1375.1
Level $1444$
Weight $1$
Character 1444.1375
Analytic conductor $0.721$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1444,1,Mod(1151,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1444.1151");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1444.g (of order \(6\), degree \(2\), 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: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.2085136.1

Embedding invariants

Embedding label 1375.1
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 1444.1375
Dual form 1444.1.g.a.1151.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.309017 - 0.535233i) q^{5} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.309017 - 0.535233i) q^{5} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.309017 + 0.535233i) q^{10} +(0.809017 - 1.40126i) q^{13} +(-0.500000 - 0.866025i) q^{16} +(0.809017 + 1.40126i) q^{17} +1.00000 q^{18} +0.618034 q^{20} +(0.309017 - 0.535233i) q^{25} -1.61803 q^{26} +(0.809017 - 1.40126i) q^{29} +(-0.500000 + 0.866025i) q^{32} +(0.809017 - 1.40126i) q^{34} +(-0.500000 - 0.866025i) q^{36} +0.618034 q^{37} +(-0.309017 - 0.535233i) q^{40} +(-0.309017 - 0.535233i) q^{41} +0.618034 q^{45} +1.00000 q^{49} -0.618034 q^{50} +(0.809017 + 1.40126i) q^{52} +(-0.309017 + 0.535233i) q^{53} -1.61803 q^{58} +(0.809017 - 1.40126i) q^{61} +1.00000 q^{64} -1.00000 q^{65} -1.61803 q^{68} +(-0.500000 + 0.866025i) q^{72} +(-0.309017 - 0.535233i) q^{73} +(-0.309017 - 0.535233i) q^{74} +(-0.309017 + 0.535233i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.309017 + 0.535233i) q^{82} +(0.500000 - 0.866025i) q^{85} +(-0.309017 + 0.535233i) q^{89} +(-0.309017 - 0.535233i) q^{90} +(0.809017 + 1.40126i) q^{97} +(-0.500000 - 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{8} - 2 q^{9} + q^{10} + q^{13} - 2 q^{16} + q^{17} + 4 q^{18} - 2 q^{20} - q^{25} - 2 q^{26} + q^{29} - 2 q^{32} + q^{34} - 2 q^{36} - 2 q^{37} + q^{40} + q^{41} - 2 q^{45} + 4 q^{49} + 2 q^{50} + q^{52} + q^{53} - 2 q^{58} + q^{61} + 4 q^{64} - 4 q^{65} - 2 q^{68} - 2 q^{72} + q^{73} + q^{74} + q^{80} - 2 q^{81} + q^{82} + 2 q^{85} + q^{89} + q^{90} + q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000 1.00000
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.500000 0.866025i
\(17\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(18\) 1.00000 1.00000
\(19\) 0 0
\(20\) 0.618034 0.618034
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0.309017 0.535233i 0.309017 0.535233i
\(26\) −1.61803 −1.61803
\(27\) 0 0
\(28\) 0 0
\(29\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(33\) 0 0
\(34\) 0.809017 1.40126i 0.809017 1.40126i
\(35\) 0 0
\(36\) −0.500000 0.866025i −0.500000 0.866025i
\(37\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.309017 0.535233i −0.309017 0.535233i
\(41\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(42\) 0 0
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) 0.618034 0.618034
\(46\) 0 0
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) −0.618034 −0.618034
\(51\) 0 0
\(52\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(53\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −1.61803 −1.61803
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) −1.00000 −1.00000
\(66\) 0 0
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) −1.61803 −1.61803
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(73\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(74\) −0.309017 0.535233i −0.309017 0.535233i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(81\) −0.500000 0.866025i −0.500000 0.866025i
\(82\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0.500000 0.866025i 0.500000 0.866025i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(90\) −0.309017 0.535233i −0.309017 0.535233i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(98\) −0.500000 0.866025i −0.500000 0.866025i
\(99\) 0 0
\(100\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(101\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0.809017 1.40126i 0.809017 1.40126i
\(105\) 0 0
\(106\) 0.618034 0.618034
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(117\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) −1.61803 −1.61803
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −1.00000
\(126\) 0 0
\(127\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) −0.500000 0.866025i −0.500000 0.866025i
\(129\) 0 0
\(130\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(137\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(138\) 0 0
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) −1.00000 −1.00000
\(146\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(147\) 0 0
\(148\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(149\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −1.61803 −1.61803
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.618034 0.618034
\(161\) 0 0
\(162\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0.618034 0.618034
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(168\) 0 0
\(169\) −0.809017 1.40126i −0.809017 1.40126i
\(170\) −1.00000 −1.00000
\(171\) 0 0
\(172\) 0 0
\(173\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.618034 0.618034
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(181\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.190983 0.330792i −0.190983 0.330792i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(194\) 0.809017 1.40126i 0.809017 1.40126i
\(195\) 0 0
\(196\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(197\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(198\) 0 0
\(199\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0.309017 0.535233i 0.309017 0.535233i
\(201\) 0 0
\(202\) 0.618034 0.618034
\(203\) 0 0
\(204\) 0 0
\(205\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(206\) 0 0
\(207\) 0 0
\(208\) −1.61803 −1.61803
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(212\) −0.309017 0.535233i −0.309017 0.535233i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(219\) 0 0
\(220\) 0 0
\(221\) 2.61803 2.61803
\(222\) 0 0
\(223\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 0 0
\(225\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(226\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.809017 1.40126i 0.809017 1.40126i
\(233\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(234\) 0.809017 1.40126i 0.809017 1.40126i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) −0.500000 0.866025i −0.500000 0.866025i
\(243\) 0 0
\(244\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(245\) −0.309017 0.535233i −0.309017 0.535233i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(251\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.500000 0.866025i 0.500000 0.866025i
\(261\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 0 0
\(265\) 0.381966 0.381966
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(270\) 0 0
\(271\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) 0.809017 1.40126i 0.809017 1.40126i
\(273\) 0 0
\(274\) 0.618034 0.618034
\(275\) 0 0
\(276\) 0 0
\(277\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(282\) 0 0
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.500000 0.866025i −0.500000 0.866025i
\(289\) −0.809017 + 1.40126i −0.809017 + 1.40126i
\(290\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(291\) 0 0
\(292\) 0.618034 0.618034
\(293\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.618034 0.618034
\(297\) 0 0
\(298\) 0.809017 1.40126i 0.809017 1.40126i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.00000 −1.00000
\(306\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(307\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(314\) 0.809017 1.40126i 0.809017 1.40126i
\(315\) 0 0
\(316\) 0 0
\(317\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.309017 0.535233i −0.309017 0.535233i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) −0.500000 0.866025i −0.500000 0.866025i
\(326\) 0 0
\(327\) 0 0
\(328\) −0.309017 0.535233i −0.309017 0.535233i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(338\) −0.809017 + 1.40126i −0.809017 + 1.40126i
\(339\) 0 0
\(340\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.309017 0.535233i −0.309017 0.535233i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0.618034 0.618034
\(361\) 0 0
\(362\) 2.00000 2.00000
\(363\) 0 0
\(364\) 0 0
\(365\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0 0
\(369\) 0.618034 0.618034
\(370\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(371\) 0 0
\(372\) 0 0
\(373\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.30902 2.26728i −1.30902 2.26728i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(387\) 0 0
\(388\) −1.61803 −1.61803
\(389\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 1.00000
\(393\) 0 0
\(394\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(395\) 0 0
\(396\) 0 0
\(397\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.618034 −0.618034
\(401\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.309017 0.535233i −0.309017 0.535233i
\(405\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(410\) 0.381966 0.381966
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(425\) 1.00000 1.00000
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.618034 0.618034
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(442\) −1.30902 2.26728i −1.30902 2.26728i
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0.381966 0.381966
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(450\) 0.309017 0.535233i 0.309017 0.535233i
\(451\) 0 0
\(452\) 0.809017 1.40126i 0.809017 1.40126i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(458\) −0.309017 0.535233i −0.309017 0.535233i
\(459\) 0 0
\(460\) 0 0
\(461\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −1.61803 −1.61803
\(465\) 0 0
\(466\) 0.809017 1.40126i 0.809017 1.40126i
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −1.61803 −1.61803
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.309017 0.535233i −0.309017 0.535233i
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0.500000 0.866025i 0.500000 0.866025i
\(482\) 2.00000 2.00000
\(483\) 0 0
\(484\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(485\) 0.500000 0.866025i 0.500000 0.866025i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0.809017 1.40126i 0.809017 1.40126i
\(489\) 0 0
\(490\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(491\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) 2.61803 2.61803
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) 0.500000 0.866025i 0.500000 0.866025i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) 0 0
\(505\) 0.381966 0.381966
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 1.00000
\(513\) 0 0
\(514\) 0.618034 0.618034
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −1.00000 −1.00000
\(521\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(522\) 0.809017 1.40126i 0.809017 1.40126i
\(523\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) −0.190983 0.330792i −0.190983 0.330792i
\(531\) 0 0
\(532\) 0 0
\(533\) −1.00000 −1.00000
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.809017 1.40126i 0.809017 1.40126i
\(539\) 0 0
\(540\) 0 0
\(541\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.61803 −1.61803
\(545\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(546\) 0 0
\(547\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) −0.309017 0.535233i −0.309017 0.535233i
\(549\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −0.309017 0.535233i −0.309017 0.535233i
\(555\) 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
\(562\) −1.61803 −1.61803
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(577\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(578\) 1.61803 1.61803
\(579\) 0 0
\(580\) 0.500000 0.866025i 0.500000 0.866025i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.309017 0.535233i −0.309017 0.535233i
\(585\) 0.500000 0.866025i 0.500000 0.866025i
\(586\) −1.00000 1.73205i −1.00000 1.73205i
\(587\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.309017 0.535233i −0.309017 0.535233i
\(593\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.61803 −1.61803
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.309017 0.535233i −0.309017 0.535233i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(611\) 0 0
\(612\) 0.809017 1.40126i 0.809017 1.40126i
\(613\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0.618034 0.618034
\(627\) 0 0
\(628\) −1.61803 −1.61803
\(629\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(630\) 0 0
\(631\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.618034 0.618034
\(635\) 0 0
\(636\) 0 0
\(637\) 0.809017 1.40126i 0.809017 1.40126i
\(638\) 0 0
\(639\) 0 0
\(640\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(641\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(642\) 0 0
\(643\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −0.500000 0.866025i −0.500000 0.866025i
\(649\) 0 0
\(650\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(651\) 0 0
\(652\) 0 0
\(653\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(657\) 0.618034 0.618034
\(658\) 0 0
\(659\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 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 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.618034 0.618034
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(674\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(675\) 0 0
\(676\) 1.61803 1.61803
\(677\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.500000 0.866025i 0.500000 0.866025i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0.381966 0.381966
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0.618034 0.618034
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.500000 0.866025i 0.500000 0.866025i
\(698\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(699\) 0 0
\(700\) 0 0
\(701\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(707\) 0 0
\(708\) 0 0
\(709\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) −0.309017 0.535233i −0.309017 0.535233i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −1.00000 1.73205i −1.00000 1.73205i
\(725\) −0.500000 0.866025i −0.500000 0.866025i
\(726\) 0 0
\(727\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0.381966 0.381966
\(731\) 0 0
\(732\) 0 0
\(733\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.309017 0.535233i −0.309017 0.535233i
\(739\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0.381966 0.381966
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(744\) 0 0
\(745\) 0.500000 0.866025i 0.500000 0.866025i
\(746\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −1.30902 + 2.26728i −1.30902 + 2.26728i
\(755\) 0 0
\(756\) 0 0
\(757\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.00000 2.00000
\(773\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(777\) 0 0
\(778\) −1.61803 −1.61803
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.500000 0.866025i −0.500000 0.866025i
\(785\) 0.500000 0.866025i 0.500000 0.866025i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0.809017 1.40126i 0.809017 1.40126i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.30902 2.26728i −1.30902 2.26728i
\(794\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(795\) 0 0
\(796\) 0 0
\(797\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(801\) −0.309017 0.535233i −0.309017 0.535233i
\(802\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(809\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(810\) 0.618034 0.618034
\(811\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0.618034 0.618034
\(819\) 0 0
\(820\) −0.190983 0.330792i −0.190983 0.330792i
\(821\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(822\) 0 0
\(823\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(828\) 0 0
\(829\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.809017 1.40126i 0.809017 1.40126i
\(833\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 0 0
\(841\) −0.809017 1.40126i −0.809017 1.40126i
\(842\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(843\) 0 0
\(844\) 0 0
\(845\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.618034 0.618034
\(849\) 0 0
\(850\) −0.500000 0.866025i −0.500000 0.866025i
\(851\) 0 0
\(852\) 0 0
\(853\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(858\) 0 0
\(859\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(866\) 0.618034 0.618034
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.309017 0.535233i −0.309017 0.535233i
\(873\) −1.61803 −1.61803
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(882\) 1.00000 1.00000
\(883\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) −1.30902 + 2.26728i −1.30902 + 2.26728i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −0.190983 0.330792i −0.190983 0.330792i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(899\) 0 0
\(900\) −0.618034 −0.618034
\(901\) −1.00000 −1.00000
\(902\) 0 0
\(903\) 0 0
\(904\) −1.61803 −1.61803
\(905\) 1.23607 1.23607
\(906\) 0 0
\(907\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(908\) 0 0
\(909\) −0.309017 0.535233i −0.309017 0.535233i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.309017 0.535233i −0.309017 0.535233i
\(915\) 0 0
\(916\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(923\) 0 0
\(924\) 0 0
\(925\) 0.190983 0.330792i 0.190983 0.330792i
\(926\) 0 0
\(927\) 0 0
\(928\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(929\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.61803 −1.61803
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(937\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 0 0
\(949\) −1.00000 −1.00000
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(954\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) −1.00000 −1.00000
\(963\) 0 0
\(964\) −1.00000 1.73205i −1.00000 1.73205i
\(965\) −0.618034 + 1.07047i −0.618034 + 1.07047i
\(966\) 0 0
\(967\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) 1.00000 1.00000
\(969\) 0 0
\(970\) −1.00000 −1.00000
\(971\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.61803 −1.61803
\(977\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.618034 0.618034
\(981\) 0.618034 0.618034
\(982\) 0 0
\(983\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(984\) 0 0
\(985\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(986\) −1.30902 2.26728i −1.30902 2.26728i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(998\) 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 1444.1.g.a.1375.1 4
4.3 odd 2 CM 1444.1.g.a.1375.1 4
19.2 odd 18 1444.1.l.b.1111.1 12
19.3 odd 18 1444.1.l.b.595.2 12
19.4 even 9 1444.1.l.a.967.1 12
19.5 even 9 1444.1.l.a.99.2 12
19.6 even 9 1444.1.l.a.423.2 12
19.7 even 3 1444.1.b.b.723.2 yes 2
19.8 odd 6 1444.1.g.b.1151.1 4
19.9 even 9 1444.1.l.a.415.2 12
19.10 odd 18 1444.1.l.b.415.2 12
19.11 even 3 inner 1444.1.g.a.1151.1 4
19.12 odd 6 1444.1.b.a.723.2 2
19.13 odd 18 1444.1.l.b.423.2 12
19.14 odd 18 1444.1.l.b.99.2 12
19.15 odd 18 1444.1.l.b.967.1 12
19.16 even 9 1444.1.l.a.595.2 12
19.17 even 9 1444.1.l.a.1111.1 12
19.18 odd 2 1444.1.g.b.1375.1 4
76.3 even 18 1444.1.l.b.595.2 12
76.7 odd 6 1444.1.b.b.723.2 yes 2
76.11 odd 6 inner 1444.1.g.a.1151.1 4
76.15 even 18 1444.1.l.b.967.1 12
76.23 odd 18 1444.1.l.a.967.1 12
76.27 even 6 1444.1.g.b.1151.1 4
76.31 even 6 1444.1.b.a.723.2 2
76.35 odd 18 1444.1.l.a.595.2 12
76.43 odd 18 1444.1.l.a.99.2 12
76.47 odd 18 1444.1.l.a.415.2 12
76.51 even 18 1444.1.l.b.423.2 12
76.55 odd 18 1444.1.l.a.1111.1 12
76.59 even 18 1444.1.l.b.1111.1 12
76.63 odd 18 1444.1.l.a.423.2 12
76.67 even 18 1444.1.l.b.415.2 12
76.71 even 18 1444.1.l.b.99.2 12
76.75 even 2 1444.1.g.b.1375.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1444.1.b.a.723.2 2 19.12 odd 6
1444.1.b.a.723.2 2 76.31 even 6
1444.1.b.b.723.2 yes 2 19.7 even 3
1444.1.b.b.723.2 yes 2 76.7 odd 6
1444.1.g.a.1151.1 4 19.11 even 3 inner
1444.1.g.a.1151.1 4 76.11 odd 6 inner
1444.1.g.a.1375.1 4 1.1 even 1 trivial
1444.1.g.a.1375.1 4 4.3 odd 2 CM
1444.1.g.b.1151.1 4 19.8 odd 6
1444.1.g.b.1151.1 4 76.27 even 6
1444.1.g.b.1375.1 4 19.18 odd 2
1444.1.g.b.1375.1 4 76.75 even 2
1444.1.l.a.99.2 12 19.5 even 9
1444.1.l.a.99.2 12 76.43 odd 18
1444.1.l.a.415.2 12 19.9 even 9
1444.1.l.a.415.2 12 76.47 odd 18
1444.1.l.a.423.2 12 19.6 even 9
1444.1.l.a.423.2 12 76.63 odd 18
1444.1.l.a.595.2 12 19.16 even 9
1444.1.l.a.595.2 12 76.35 odd 18
1444.1.l.a.967.1 12 19.4 even 9
1444.1.l.a.967.1 12 76.23 odd 18
1444.1.l.a.1111.1 12 19.17 even 9
1444.1.l.a.1111.1 12 76.55 odd 18
1444.1.l.b.99.2 12 19.14 odd 18
1444.1.l.b.99.2 12 76.71 even 18
1444.1.l.b.415.2 12 19.10 odd 18
1444.1.l.b.415.2 12 76.67 even 18
1444.1.l.b.423.2 12 19.13 odd 18
1444.1.l.b.423.2 12 76.51 even 18
1444.1.l.b.595.2 12 19.3 odd 18
1444.1.l.b.595.2 12 76.3 even 18
1444.1.l.b.967.1 12 19.15 odd 18
1444.1.l.b.967.1 12 76.15 even 18
1444.1.l.b.1111.1 12 19.2 odd 18
1444.1.l.b.1111.1 12 76.59 even 18