Properties

Label 1600.1.bd.a.991.1
Level $1600$
Weight $1$
Character 1600.991
Analytic conductor $0.799$
Analytic rank $0$
Dimension $8$
Projective image $A_{5}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,1,Mod(31,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.31");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1600.bd (of order \(10\), degree \(4\), 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: \(0.798504020213\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{5}\)
Projective field: Galois closure of 5.1.25000000.3

Embedding invariants

Embedding label 991.1
Root \(-0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 1600.991
Dual form 1600.1.bd.a.1311.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.190983 - 0.587785i) q^{3} +(-0.951057 + 0.309017i) q^{5} +1.00000i q^{7} +(0.500000 - 0.363271i) q^{9} +O(q^{10})\) \(q+(-0.190983 - 0.587785i) q^{3} +(-0.951057 + 0.309017i) q^{5} +1.00000i q^{7} +(0.500000 - 0.363271i) q^{9} +(0.809017 + 0.587785i) q^{11} +(-0.587785 - 0.809017i) q^{13} +(0.363271 + 0.500000i) q^{15} +(-0.190983 + 0.587785i) q^{17} +(-0.500000 + 1.53884i) q^{19} +(0.587785 - 0.190983i) q^{21} +(-0.587785 + 0.809017i) q^{23} +(0.809017 - 0.587785i) q^{25} +(-0.809017 - 0.587785i) q^{27} +(0.951057 - 0.309017i) q^{29} +(0.951057 + 0.309017i) q^{31} +(0.190983 - 0.587785i) q^{33} +(-0.309017 - 0.951057i) q^{35} +(0.951057 + 1.30902i) q^{37} +(-0.363271 + 0.500000i) q^{39} +(-0.809017 + 0.587785i) q^{41} +1.00000 q^{43} +(-0.363271 + 0.500000i) q^{45} +(0.951057 - 0.309017i) q^{47} +0.381966 q^{51} +(-0.951057 - 0.309017i) q^{55} +1.00000 q^{57} +(0.363271 - 0.500000i) q^{61} +(0.363271 + 0.500000i) q^{63} +(0.809017 + 0.587785i) q^{65} +(-0.309017 + 0.951057i) q^{67} +(0.587785 + 0.190983i) q^{69} +(-0.951057 + 0.309017i) q^{71} +(-0.500000 - 0.363271i) q^{75} +(-0.587785 + 0.809017i) q^{77} +(0.309017 - 0.951057i) q^{83} -0.618034i q^{85} +(-0.363271 - 0.500000i) q^{87} +(0.809017 - 0.587785i) q^{91} -0.618034i q^{93} -1.61803i q^{95} +(-0.500000 - 1.53884i) q^{97} +0.618034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} + 4 q^{9} + 2 q^{11} - 6 q^{17} - 4 q^{19} + 2 q^{25} - 2 q^{27} + 6 q^{33} + 2 q^{35} - 2 q^{41} + 8 q^{43} + 12 q^{51} + 8 q^{57} + 2 q^{65} + 2 q^{67} - 4 q^{75} - 2 q^{83} + 2 q^{91} - 4 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(6\) 0 0
\(7\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 0.500000 0.363271i 0.500000 0.363271i
\(10\) 0 0
\(11\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(12\) 0 0
\(13\) −0.587785 0.809017i −0.587785 0.809017i 0.406737 0.913545i \(-0.366667\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(14\) 0 0
\(15\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(16\) 0 0
\(17\) −0.190983 + 0.587785i −0.190983 + 0.587785i 0.809017 + 0.587785i \(0.200000\pi\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(20\) 0 0
\(21\) 0.587785 0.190983i 0.587785 0.190983i
\(22\) 0 0
\(23\) −0.587785 + 0.809017i −0.587785 + 0.809017i −0.994522 0.104528i \(-0.966667\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(24\) 0 0
\(25\) 0.809017 0.587785i 0.809017 0.587785i
\(26\) 0 0
\(27\) −0.809017 0.587785i −0.809017 0.587785i
\(28\) 0 0
\(29\) 0.951057 0.309017i 0.951057 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(30\) 0 0
\(31\) 0.951057 + 0.309017i 0.951057 + 0.309017i 0.743145 0.669131i \(-0.233333\pi\)
0.207912 + 0.978148i \(0.433333\pi\)
\(32\) 0 0
\(33\) 0.190983 0.587785i 0.190983 0.587785i
\(34\) 0 0
\(35\) −0.309017 0.951057i −0.309017 0.951057i
\(36\) 0 0
\(37\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(40\) 0 0
\(41\) −0.809017 + 0.587785i −0.809017 + 0.587785i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(42\) 0 0
\(43\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0 0
\(45\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(46\) 0 0
\(47\) 0.951057 0.309017i 0.951057 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.381966 0.381966
\(52\) 0 0
\(53\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(54\) 0 0
\(55\) −0.951057 0.309017i −0.951057 0.309017i
\(56\) 0 0
\(57\) 1.00000 1.00000
\(58\) 0 0
\(59\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(60\) 0 0
\(61\) 0.363271 0.500000i 0.363271 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(62\) 0 0
\(63\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(64\) 0 0
\(65\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(66\) 0 0
\(67\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(68\) 0 0
\(69\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(70\) 0 0
\(71\) −0.951057 + 0.309017i −0.951057 + 0.309017i −0.743145 0.669131i \(-0.766667\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(72\) 0 0
\(73\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(74\) 0 0
\(75\) −0.500000 0.363271i −0.500000 0.363271i
\(76\) 0 0
\(77\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(78\) 0 0
\(79\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.309017 0.951057i 0.309017 0.951057i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(84\) 0 0
\(85\) 0.618034i 0.618034i
\(86\) 0 0
\(87\) −0.363271 0.500000i −0.363271 0.500000i
\(88\) 0 0
\(89\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(90\) 0 0
\(91\) 0.809017 0.587785i 0.809017 0.587785i
\(92\) 0 0
\(93\) 0.618034i 0.618034i
\(94\) 0 0
\(95\) 1.61803i 1.61803i
\(96\) 0 0
\(97\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(98\) 0 0
\(99\) 0.618034 0.618034
\(100\) 0 0
\(101\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(102\) 0 0
\(103\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(104\) 0 0
\(105\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(106\) 0 0
\(107\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(108\) 0 0
\(109\) 0.587785 + 0.809017i 0.587785 + 0.809017i 0.994522 0.104528i \(-0.0333333\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(110\) 0 0
\(111\) 0.587785 0.809017i 0.587785 0.809017i
\(112\) 0 0
\(113\) −0.809017 + 0.587785i −0.809017 + 0.587785i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(114\) 0 0
\(115\) 0.309017 0.951057i 0.309017 0.951057i
\(116\) 0 0
\(117\) −0.587785 0.190983i −0.587785 0.190983i
\(118\) 0 0
\(119\) −0.587785 0.190983i −0.587785 0.190983i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(124\) 0 0
\(125\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(126\) 0 0
\(127\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(128\) 0 0
\(129\) −0.190983 0.587785i −0.190983 0.587785i
\(130\) 0 0
\(131\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(132\) 0 0
\(133\) −1.53884 0.500000i −1.53884 0.500000i
\(134\) 0 0
\(135\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(136\) 0 0
\(137\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(140\) 0 0
\(141\) −0.363271 0.500000i −0.363271 0.500000i
\(142\) 0 0
\(143\) 1.00000i 1.00000i
\(144\) 0 0
\(145\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0.118034 + 0.363271i 0.118034 + 0.363271i
\(154\) 0 0
\(155\) −1.00000 −1.00000
\(156\) 0 0
\(157\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.809017 0.587785i −0.809017 0.587785i
\(162\) 0 0
\(163\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(164\) 0 0
\(165\) 0.618034i 0.618034i
\(166\) 0 0
\(167\) −1.53884 0.500000i −1.53884 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(172\) 0 0
\(173\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(174\) 0 0
\(175\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.309017 + 0.951057i 0.309017 + 0.951057i 0.978148 + 0.207912i \(0.0666667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(180\) 0 0
\(181\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(182\) 0 0
\(183\) −0.363271 0.118034i −0.363271 0.118034i
\(184\) 0 0
\(185\) −1.30902 0.951057i −1.30902 0.951057i
\(186\) 0 0
\(187\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(188\) 0 0
\(189\) 0.587785 0.809017i 0.587785 0.809017i
\(190\) 0 0
\(191\) −0.587785 0.809017i −0.587785 0.809017i 0.406737 0.913545i \(-0.366667\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0.190983 0.587785i 0.190983 0.587785i
\(196\) 0 0
\(197\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(198\) 0 0
\(199\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(200\) 0 0
\(201\) 0.618034 0.618034
\(202\) 0 0
\(203\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(204\) 0 0
\(205\) 0.587785 0.809017i 0.587785 0.809017i
\(206\) 0 0
\(207\) 0.618034i 0.618034i
\(208\) 0 0
\(209\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(210\) 0 0
\(211\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(212\) 0 0
\(213\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(214\) 0 0
\(215\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(216\) 0 0
\(217\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.587785 0.190983i 0.587785 0.190983i
\(222\) 0 0
\(223\) 0.587785 0.809017i 0.587785 0.809017i −0.406737 0.913545i \(-0.633333\pi\)
0.994522 + 0.104528i \(0.0333333\pi\)
\(224\) 0 0
\(225\) 0.190983 0.587785i 0.190983 0.587785i
\(226\) 0 0
\(227\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(228\) 0 0
\(229\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(230\) 0 0
\(231\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(232\) 0 0
\(233\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(234\) 0 0
\(235\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.951057 1.30902i 0.951057 1.30902i 1.00000i \(-0.5\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(240\) 0 0
\(241\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(242\) 0 0
\(243\) −1.00000 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.53884 0.500000i 1.53884 0.500000i
\(248\) 0 0
\(249\) −0.618034 −0.618034
\(250\) 0 0
\(251\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(252\) 0 0
\(253\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(254\) 0 0
\(255\) −0.363271 + 0.118034i −0.363271 + 0.118034i
\(256\) 0 0
\(257\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(258\) 0 0
\(259\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(260\) 0 0
\(261\) 0.363271 0.500000i 0.363271 0.500000i
\(262\) 0 0
\(263\) −0.951057 1.30902i −0.951057 1.30902i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(-0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.587785 0.190983i −0.587785 0.190983i 1.00000i \(-0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(270\) 0 0
\(271\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(272\) 0 0
\(273\) −0.500000 0.363271i −0.500000 0.363271i
\(274\) 0 0
\(275\) 1.00000 1.00000
\(276\) 0 0
\(277\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(278\) 0 0
\(279\) 0.587785 0.190983i 0.587785 0.190983i
\(280\) 0 0
\(281\) 0.309017 0.951057i 0.309017 0.951057i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(282\) 0 0
\(283\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(284\) 0 0
\(285\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(286\) 0 0
\(287\) −0.587785 0.809017i −0.587785 0.809017i
\(288\) 0 0
\(289\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(290\) 0 0
\(291\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(292\) 0 0
\(293\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.309017 0.951057i −0.309017 0.951057i
\(298\) 0 0
\(299\) 1.00000 1.00000
\(300\) 0 0
\(301\) 1.00000i 1.00000i
\(302\) 0 0
\(303\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(304\) 0 0
\(305\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −0.224514 0.309017i −0.224514 0.309017i
\(310\) 0 0
\(311\) 0.363271 0.500000i 0.363271 0.500000i −0.587785 0.809017i \(-0.700000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(312\) 0 0
\(313\) 0.500000 0.363271i 0.500000 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(314\) 0 0
\(315\) −0.500000 0.363271i −0.500000 0.363271i
\(316\) 0 0
\(317\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(320\) 0 0
\(321\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(322\) 0 0
\(323\) −0.809017 0.587785i −0.809017 0.587785i
\(324\) 0 0
\(325\) −0.951057 0.309017i −0.951057 0.309017i
\(326\) 0 0
\(327\) 0.363271 0.500000i 0.363271 0.500000i
\(328\) 0 0
\(329\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(330\) 0 0
\(331\) 0.500000 1.53884i 0.500000 1.53884i −0.309017 0.951057i \(-0.600000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(332\) 0 0
\(333\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(334\) 0 0
\(335\) 1.00000i 1.00000i
\(336\) 0 0
\(337\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(338\) 0 0
\(339\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(340\) 0 0
\(341\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(342\) 0 0
\(343\) 1.00000i 1.00000i
\(344\) 0 0
\(345\) −0.618034 −0.618034
\(346\) 0 0
\(347\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(348\) 0 0
\(349\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(350\) 0 0
\(351\) 1.00000i 1.00000i
\(352\) 0 0
\(353\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(354\) 0 0
\(355\) 0.809017 0.587785i 0.809017 0.587785i
\(356\) 0 0
\(357\) 0.381966i 0.381966i
\(358\) 0 0
\(359\) 0.363271 + 0.500000i 0.363271 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(360\) 0 0
\(361\) −1.30902 0.951057i −1.30902 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.951057 0.309017i −0.951057 0.309017i −0.207912 0.978148i \(-0.566667\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(368\) 0 0
\(369\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.587785 0.809017i 0.587785 0.809017i −0.406737 0.913545i \(-0.633333\pi\)
0.994522 + 0.104528i \(0.0333333\pi\)
\(374\) 0 0
\(375\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(376\) 0 0
\(377\) −0.809017 0.587785i −0.809017 0.587785i
\(378\) 0 0
\(379\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0.309017 0.951057i 0.309017 0.951057i
\(386\) 0 0
\(387\) 0.500000 0.363271i 0.500000 0.363271i
\(388\) 0 0
\(389\) 0.951057 1.30902i 0.951057 1.30902i 1.00000i \(-0.5\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(390\) 0 0
\(391\) −0.363271 0.500000i −0.363271 0.500000i
\(392\) 0 0
\(393\) 0.618034 0.618034
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.951057 0.309017i 0.951057 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(398\) 0 0
\(399\) 1.00000i 1.00000i
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −0.309017 0.951057i −0.309017 0.951057i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.61803i 1.61803i
\(408\) 0 0
\(409\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(410\) 0 0
\(411\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.00000i 1.00000i
\(416\) 0 0
\(417\) −0.118034 + 0.363271i −0.118034 + 0.363271i
\(418\) 0 0
\(419\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(420\) 0 0
\(421\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(422\) 0 0
\(423\) 0.363271 0.500000i 0.363271 0.500000i
\(424\) 0 0
\(425\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(426\) 0 0
\(427\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(428\) 0 0
\(429\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(430\) 0 0
\(431\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(432\) 0 0
\(433\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(434\) 0 0
\(435\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(436\) 0 0
\(437\) −0.951057 1.30902i −0.951057 1.30902i
\(438\) 0 0
\(439\) −0.587785 + 0.809017i −0.587785 + 0.809017i −0.994522 0.104528i \(-0.966667\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.587785 0.190983i 0.587785 0.190983i
\(448\) 0 0
\(449\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(450\) 0 0
\(451\) −1.00000 −1.00000
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(456\) 0 0
\(457\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(458\) 0 0
\(459\) 0.500000 0.363271i 0.500000 0.363271i
\(460\) 0 0
\(461\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(462\) 0 0
\(463\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(466\) 0 0
\(467\) 0.309017 0.951057i 0.309017 0.951057i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(468\) 0 0
\(469\) −0.951057 0.309017i −0.951057 0.309017i
\(470\) 0 0
\(471\) 0.363271 0.118034i 0.363271 0.118034i
\(472\) 0 0
\(473\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(474\) 0 0
\(475\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(480\) 0 0
\(481\) 0.500000 1.53884i 0.500000 1.53884i
\(482\) 0 0
\(483\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(484\) 0 0
\(485\) 0.951057 + 1.30902i 0.951057 + 1.30902i
\(486\) 0 0
\(487\) −0.587785 0.809017i −0.587785 0.809017i 0.406737 0.913545i \(-0.366667\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(488\) 0 0
\(489\) −0.809017 0.587785i −0.809017 0.587785i
\(490\) 0 0
\(491\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(492\) 0 0
\(493\) 0.618034i 0.618034i
\(494\) 0 0
\(495\) −0.587785 + 0.190983i −0.587785 + 0.190983i
\(496\) 0 0
\(497\) −0.309017 0.951057i −0.309017 0.951057i
\(498\) 0 0
\(499\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(500\) 0 0
\(501\) 1.00000i 1.00000i
\(502\) 0 0
\(503\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(504\) 0 0
\(505\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.30902 0.951057i 1.30902 0.951057i
\(514\) 0 0
\(515\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(516\) 0 0
\(517\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(522\) 0 0
\(523\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(524\) 0 0
\(525\) 0.363271 0.500000i 0.363271 0.500000i
\(526\) 0 0
\(527\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(534\) 0 0
\(535\) 1.53884 0.500000i 1.53884 0.500000i
\(536\) 0 0
\(537\) 0.500000 0.363271i 0.500000 0.363271i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.587785 0.809017i −0.587785 0.809017i 0.406737 0.913545i \(-0.366667\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.809017 0.587785i −0.809017 0.587785i
\(546\) 0 0
\(547\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(548\) 0 0
\(549\) 0.381966i 0.381966i
\(550\) 0 0
\(551\) 1.61803i 1.61803i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(556\) 0 0
\(557\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(558\) 0 0
\(559\) −0.587785 0.809017i −0.587785 0.809017i
\(560\) 0 0
\(561\) 0.309017 + 0.224514i 0.309017 + 0.224514i
\(562\) 0 0
\(563\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0.587785 0.809017i 0.587785 0.809017i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.309017 0.951057i 0.309017 0.951057i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(570\) 0 0
\(571\) −0.190983 0.587785i −0.190983 0.587785i 0.809017 0.587785i \(-0.200000\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(574\) 0 0
\(575\) 1.00000i 1.00000i
\(576\) 0 0
\(577\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.618034 0.618034
\(586\) 0 0
\(587\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(588\) 0 0
\(589\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0.618034 0.618034
\(596\) 0 0
\(597\) −0.363271 + 0.118034i −0.363271 + 0.118034i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(608\) 0 0
\(609\) 0.500000 0.363271i 0.500000 0.363271i
\(610\) 0 0
\(611\) −0.809017 0.587785i −0.809017 0.587785i
\(612\) 0 0
\(613\) −0.363271 0.500000i −0.363271 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(614\) 0 0
\(615\) −0.587785 0.190983i −0.587785 0.190983i
\(616\) 0 0
\(617\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(618\) 0 0
\(619\) 0.309017 0.951057i 0.309017 0.951057i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(620\) 0 0
\(621\) 0.951057 0.309017i 0.951057 0.309017i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.309017 0.951057i 0.309017 0.951057i
\(626\) 0 0
\(627\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(628\) 0 0
\(629\) −0.951057 + 0.309017i −0.951057 + 0.309017i
\(630\) 0 0
\(631\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(632\) 0 0
\(633\) −0.118034 + 0.363271i −0.118034 + 0.363271i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(640\) 0 0
\(641\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(642\) 0 0
\(643\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 0 0
\(645\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(646\) 0 0
\(647\) 0.951057 0.309017i 0.951057 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.618034 0.618034
\(652\) 0 0
\(653\) 0.951057 0.309017i 0.951057 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(654\) 0 0
\(655\) 1.00000i 1.00000i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(660\) 0 0
\(661\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(662\) 0 0
\(663\) −0.224514 0.309017i −0.224514 0.309017i
\(664\) 0 0
\(665\) 1.61803 1.61803
\(666\) 0 0
\(667\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(668\) 0 0
\(669\) −0.587785 0.190983i −0.587785 0.190983i
\(670\) 0 0
\(671\) 0.587785 0.190983i 0.587785 0.190983i
\(672\) 0 0
\(673\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(674\) 0 0
\(675\) −1.00000 −1.00000
\(676\) 0 0
\(677\) −0.363271 + 0.500000i −0.363271 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(678\) 0 0
\(679\) 1.53884 0.500000i 1.53884 0.500000i
\(680\) 0 0
\(681\) 0.190983 0.587785i 0.190983 0.587785i
\(682\) 0 0
\(683\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(684\) 0 0
\(685\) 0.951057 1.30902i 0.951057 1.30902i
\(686\) 0 0
\(687\) −0.224514 0.309017i −0.224514 0.309017i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(692\) 0 0
\(693\) 0.618034i 0.618034i
\(694\) 0 0
\(695\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(696\) 0 0
\(697\) −0.190983 0.587785i −0.190983 0.587785i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −2.48990 + 0.809017i −2.48990 + 0.809017i
\(704\) 0 0
\(705\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(706\) 0 0
\(707\) 1.61803 1.61803
\(708\) 0 0
\(709\) 0.587785 + 0.809017i 0.587785 + 0.809017i 0.994522 0.104528i \(-0.0333333\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(714\) 0 0
\(715\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(716\) 0 0
\(717\) −0.951057 0.309017i −0.951057 0.309017i
\(718\) 0 0
\(719\) 0.587785 + 0.190983i 0.587785 + 0.190983i 0.587785 0.809017i \(-0.300000\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(722\) 0 0
\(723\) 0.309017 + 0.224514i 0.309017 + 0.224514i
\(724\) 0 0
\(725\) 0.587785 0.809017i 0.587785 0.809017i
\(726\) 0 0
\(727\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(728\) 0 0
\(729\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(730\) 0 0
\(731\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(732\) 0 0
\(733\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(738\) 0 0
\(739\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(740\) 0 0
\(741\) −0.587785 0.809017i −0.587785 0.809017i
\(742\) 0 0
\(743\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(744\) 0 0
\(745\) −0.309017 0.951057i −0.309017 0.951057i
\(746\) 0 0
\(747\) −0.190983 0.587785i −0.190983 0.587785i
\(748\) 0 0
\(749\) 1.61803i 1.61803i
\(750\) 0 0
\(751\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(752\) 0 0
\(753\) −0.190983 0.587785i −0.190983 0.587785i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(758\) 0 0
\(759\) 0.363271 + 0.500000i 0.363271 + 0.500000i
\(760\) 0 0
\(761\) 0.500000 + 0.363271i 0.500000 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(762\) 0 0
\(763\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(764\) 0 0
\(765\) −0.224514 0.309017i −0.224514 0.309017i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(770\) 0 0
\(771\) −0.309017 0.951057i −0.309017 0.951057i
\(772\) 0 0
\(773\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(774\) 0 0
\(775\) 0.951057 0.309017i 0.951057 0.309017i
\(776\) 0 0
\(777\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(778\) 0 0
\(779\) −0.500000 1.53884i −0.500000 1.53884i
\(780\) 0 0
\(781\) −0.951057 0.309017i −0.951057 0.309017i
\(782\) 0 0
\(783\) −0.951057 0.309017i −0.951057 0.309017i
\(784\) 0 0
\(785\) −0.190983 0.587785i −0.190983 0.587785i
\(786\) 0 0
\(787\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(788\) 0 0
\(789\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(790\) 0 0
\(791\) −0.587785 0.809017i −0.587785 0.809017i
\(792\) 0 0
\(793\) −0.618034 −0.618034
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −0.587785 + 0.190983i −0.587785 + 0.190983i −0.587785 0.809017i \(-0.700000\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0.618034i 0.618034i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(806\) 0 0
\(807\) 0.381966i 0.381966i
\(808\) 0 0
\(809\) −0.809017 + 0.587785i −0.809017 + 0.587785i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(810\) 0 0
\(811\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(812\) 0 0
\(813\) 0.587785 + 0.809017i 0.587785 + 0.809017i
\(814\) 0 0
\(815\) −0.951057 + 1.30902i −0.951057 + 1.30902i
\(816\) 0 0
\(817\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(818\) 0 0
\(819\) 0.190983 0.587785i 0.190983 0.587785i
\(820\) 0 0
\(821\) −1.53884 + 0.500000i −1.53884 + 0.500000i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(822\) 0 0
\(823\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(824\) 0 0
\(825\) −0.190983 0.587785i −0.190983 0.587785i
\(826\) 0 0
\(827\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(828\) 0 0
\(829\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1.61803 1.61803
\(836\) 0 0
\(837\) −0.587785 0.809017i −0.587785 0.809017i
\(838\) 0 0
\(839\) 0.587785 0.809017i 0.587785 0.809017i −0.406737 0.913545i \(-0.633333\pi\)
0.994522 + 0.104528i \(0.0333333\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0 0
\(843\) −0.618034 −0.618034
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.61803 −1.61803
\(852\) 0 0
\(853\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(854\) 0 0
\(855\) −0.587785 0.809017i −0.587785 0.809017i
\(856\) 0 0
\(857\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(858\) 0 0
\(859\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(860\) 0 0
\(861\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(862\) 0 0
\(863\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.118034 0.363271i 0.118034 0.363271i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0.951057 0.309017i 0.951057 0.309017i
\(872\) 0 0
\(873\) −0.809017 0.587785i −0.809017 0.587785i
\(874\) 0 0
\(875\) −0.809017 0.587785i −0.809017 0.587785i
\(876\) 0 0
\(877\) 0.951057 1.30902i 0.951057 1.30902i 1.00000i \(-0.5\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(878\) 0 0
\(879\) −0.363271 + 0.118034i −0.363271 + 0.118034i
\(880\) 0 0
\(881\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(882\) 0 0
\(883\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.587785 + 0.809017i 0.587785 + 0.809017i 0.994522 0.104528i \(-0.0333333\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.61803i 1.61803i
\(894\) 0 0
\(895\) −0.587785 0.809017i −0.587785 0.809017i
\(896\) 0 0
\(897\) −0.190983 0.587785i −0.190983 0.587785i
\(898\) 0 0
\(899\) 1.00000 1.00000
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0.587785 0.190983i 0.587785 0.190983i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(908\) 0 0
\(909\) −0.587785 0.809017i −0.587785 0.809017i
\(910\) 0 0
\(911\) 0.587785 0.809017i 0.587785 0.809017i −0.406737 0.913545i \(-0.633333\pi\)
0.994522 + 0.104528i \(0.0333333\pi\)
\(912\) 0 0
\(913\) 0.809017 0.587785i 0.809017 0.587785i
\(914\) 0 0
\(915\) 0.381966 0.381966
\(916\) 0 0
\(917\) −0.951057 0.309017i −0.951057 0.309017i
\(918\) 0 0
\(919\) −0.951057 0.309017i −0.951057 0.309017i −0.207912 0.978148i \(-0.566667\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(924\) 0 0
\(925\) 1.53884 + 0.500000i 1.53884 + 0.500000i
\(926\) 0 0
\(927\) 0.224514 0.309017i 0.224514 0.309017i
\(928\) 0 0
\(929\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −0.363271 0.118034i −0.363271 0.118034i
\(934\) 0 0
\(935\) 0.363271 0.500000i 0.363271 0.500000i
\(936\) 0 0
\(937\) −1.30902 + 0.951057i −1.30902 + 0.951057i −0.309017 + 0.951057i \(0.600000\pi\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) −0.309017 0.224514i −0.309017 0.224514i
\(940\) 0 0
\(941\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 1.00000i 1.00000i
\(944\) 0 0
\(945\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(946\) 0 0
\(947\) 0.500000 + 1.53884i 0.500000 + 1.53884i 0.809017 + 0.587785i \(0.200000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0.381966i 0.381966i
\(952\) 0 0
\(953\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(954\) 0 0
\(955\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(956\) 0 0
\(957\) 0.618034i 0.618034i
\(958\) 0 0
\(959\) −0.951057 1.30902i −0.951057 1.30902i
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(968\) 0 0
\(969\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(970\) 0 0
\(971\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(972\) 0 0
\(973\) 0.363271 0.500000i 0.363271 0.500000i
\(974\) 0 0
\(975\) 0.618034i 0.618034i
\(976\) 0 0
\(977\) −0.809017 0.587785i −0.809017 0.587785i 0.104528 0.994522i \(-0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(982\) 0 0
\(983\) 0.951057 + 0.309017i 0.951057 + 0.309017i 0.743145 0.669131i \(-0.233333\pi\)
0.207912 + 0.978148i \(0.433333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0.500000 0.363271i 0.500000 0.363271i
\(988\) 0 0
\(989\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(990\) 0 0
\(991\) −0.951057 1.30902i −0.951057 1.30902i −0.951057 0.309017i \(-0.900000\pi\)
1.00000i \(-0.5\pi\)
\(992\) 0 0
\(993\) −1.00000 −1.00000
\(994\) 0 0
\(995\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(996\) 0 0
\(997\) 0.951057 0.309017i 0.951057 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
0.743145 + 0.669131i \(0.233333\pi\)
\(998\) 0 0
\(999\) 1.61803i 1.61803i
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 1600.1.bd.a.991.1 8
4.3 odd 2 1600.1.bd.b.991.1 yes 8
8.3 odd 2 inner 1600.1.bd.a.991.2 yes 8
8.5 even 2 1600.1.bd.b.991.2 yes 8
25.11 even 5 inner 1600.1.bd.a.1311.2 yes 8
100.11 odd 10 1600.1.bd.b.1311.2 yes 8
200.11 odd 10 inner 1600.1.bd.a.1311.1 yes 8
200.61 even 10 1600.1.bd.b.1311.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.1.bd.a.991.1 8 1.1 even 1 trivial
1600.1.bd.a.991.2 yes 8 8.3 odd 2 inner
1600.1.bd.a.1311.1 yes 8 200.11 odd 10 inner
1600.1.bd.a.1311.2 yes 8 25.11 even 5 inner
1600.1.bd.b.991.1 yes 8 4.3 odd 2
1600.1.bd.b.991.2 yes 8 8.5 even 2
1600.1.bd.b.1311.1 yes 8 200.61 even 10
1600.1.bd.b.1311.2 yes 8 100.11 odd 10