Properties

Label 93.1.l.a.8.1
Level $93$
Weight $1$
Character 93.8
Analytic conductor $0.046$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [93,1,Mod(2,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.2");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 93.l (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: \(0.0464130461749\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.8311689.1

Embedding invariants

Embedding label 8.1
Root \(-0.309017 - 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 93.8
Dual form 93.1.l.a.35.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.309017 + 0.951057i) q^{3} +(-0.809017 - 0.587785i) q^{4} +(-0.500000 - 0.363271i) q^{7} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\) \(q+(0.309017 + 0.951057i) q^{3} +(-0.809017 - 0.587785i) q^{4} +(-0.500000 - 0.363271i) q^{7} +(-0.809017 + 0.587785i) q^{9} +(0.309017 - 0.951057i) q^{12} +(-0.500000 - 1.53884i) q^{13} +(0.309017 + 0.951057i) q^{16} +(-0.500000 + 1.53884i) q^{19} +(0.190983 - 0.587785i) q^{21} +1.00000 q^{25} +(-0.809017 - 0.587785i) q^{27} +(0.190983 + 0.587785i) q^{28} +(0.309017 + 0.951057i) q^{31} +1.00000 q^{36} +0.618034 q^{37} +(1.30902 - 0.951057i) q^{39} +(0.190983 - 0.587785i) q^{43} +(-0.809017 + 0.587785i) q^{48} +(-0.190983 - 0.587785i) q^{49} +(-0.500000 + 1.53884i) q^{52} -1.61803 q^{57} -1.61803 q^{61} +0.618034 q^{63} +(0.309017 - 0.951057i) q^{64} -1.61803 q^{67} +(-0.500000 - 0.363271i) q^{73} +(0.309017 + 0.951057i) q^{75} +(1.30902 - 0.951057i) q^{76} +(1.30902 - 0.951057i) q^{79} +(0.309017 - 0.951057i) q^{81} +(-0.500000 + 0.363271i) q^{84} +(-0.309017 + 0.951057i) q^{91} +(-0.809017 + 0.587785i) q^{93} +(1.30902 + 0.951057i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} - 2 q^{13} - q^{16} - 2 q^{19} + 3 q^{21} + 4 q^{25} - q^{27} + 3 q^{28} - q^{31} + 4 q^{36} - 2 q^{37} + 3 q^{39} + 3 q^{43} - q^{48} - 3 q^{49} - 2 q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} + 3 q^{76} + 3 q^{79} - q^{81} - 2 q^{84} + q^{91} - q^{93} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(34\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{5}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(3\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(4\) −0.809017 0.587785i −0.809017 0.587785i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(8\) 0 0
\(9\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(10\) 0 0
\(11\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(12\) 0.309017 0.951057i 0.309017 0.951057i
\(13\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(17\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(18\) 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
\(21\) 0.190983 0.587785i 0.190983 0.587785i
\(22\) 0 0
\(23\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(24\) 0 0
\(25\) 1.00000 1.00000
\(26\) 0 0
\(27\) −0.809017 0.587785i −0.809017 0.587785i
\(28\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(29\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(30\) 0 0
\(31\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(38\) 0 0
\(39\) 1.30902 0.951057i 1.30902 0.951057i
\(40\) 0 0
\(41\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(42\) 0 0
\(43\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(48\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(49\) −0.190983 0.587785i −0.190983 0.587785i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(53\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.61803 −1.61803
\(58\) 0 0
\(59\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) 0 0
\(61\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(62\) 0 0
\(63\) 0.618034 0.618034
\(64\) 0.309017 0.951057i 0.309017 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) 0 0
\(73\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(74\) 0 0
\(75\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(76\) 1.30902 0.951057i 1.30902 0.951057i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(80\) 0 0
\(81\) 0.309017 0.951057i 0.309017 0.951057i
\(82\) 0 0
\(83\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(84\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(90\) 0 0
\(91\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(92\) 0 0
\(93\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.809017 0.587785i −0.809017 0.587785i
\(101\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(102\) 0 0
\(103\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(108\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(109\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(110\) 0 0
\(111\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(112\) 0.190983 0.587785i 0.190983 0.587785i
\(113\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(122\) 0 0
\(123\) 0 0
\(124\) 0.309017 0.951057i 0.309017 0.951057i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(128\) 0 0
\(129\) 0.618034 0.618034
\(130\) 0 0
\(131\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(132\) 0 0
\(133\) 0.809017 0.587785i 0.809017 0.587785i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(138\) 0 0
\(139\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.809017 0.587785i −0.809017 0.587785i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.500000 0.363271i 0.500000 0.363271i
\(148\) −0.500000 0.363271i −0.500000 0.363271i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.61803 −1.61803
\(157\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(168\) 0 0
\(169\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(170\) 0 0
\(171\) −0.500000 1.53884i −0.500000 1.53884i
\(172\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(173\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(174\) 0 0
\(175\) −0.500000 0.363271i −0.500000 0.363271i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(180\) 0 0
\(181\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(182\) 0 0
\(183\) −0.500000 1.53884i −0.500000 1.53884i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 1.00000 1.00000
\(193\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(197\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(198\) 0 0
\(199\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(200\) 0 0
\(201\) −0.500000 1.53884i −0.500000 1.53884i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.30902 0.951057i 1.30902 0.951057i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.190983 0.587785i 0.190983 0.587785i
\(218\) 0 0
\(219\) 0.190983 0.587785i 0.190983 0.587785i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(224\) 0 0
\(225\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(226\) 0 0
\(227\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(228\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(229\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(238\) 0 0
\(239\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(240\) 0 0
\(241\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(242\) 0 0
\(243\) 1.00000 1.00000
\(244\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(245\) 0 0
\(246\) 0 0
\(247\) 2.61803 2.61803
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(252\) −0.500000 0.363271i −0.500000 0.363271i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(257\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(258\) 0 0
\(259\) −0.309017 0.224514i −0.309017 0.224514i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(269\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(270\) 0 0
\(271\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(272\) 0 0
\(273\) −1.00000 −1.00000
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(278\) 0 0
\(279\) −0.809017 0.587785i −0.809017 0.587785i
\(280\) 0 0
\(281\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(282\) 0 0
\(283\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.309017 0.951057i 0.309017 0.951057i
\(290\) 0 0
\(291\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(292\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(293\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.309017 0.951057i 0.309017 0.951057i
\(301\) −0.309017 + 0.224514i −0.309017 + 0.224514i
\(302\) 0 0
\(303\) 0 0
\(304\) −1.61803 −1.61803
\(305\) 0 0
\(306\) 0 0
\(307\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(308\) 0 0
\(309\) 0.618034 0.618034
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.61803 −1.61803
\(317\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(325\) −0.500000 1.53884i −0.500000 1.53884i
\(326\) 0 0
\(327\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(332\) 0 0
\(333\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(334\) 0 0
\(335\) 0 0
\(336\) 0.618034 0.618034
\(337\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(350\) 0 0
\(351\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(352\) 0 0
\(353\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(360\) 0 0
\(361\) −1.30902 0.951057i −1.30902 0.951057i
\(362\) 0 0
\(363\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(364\) 0.809017 0.587785i 0.809017 0.587785i
\(365\) 0 0
\(366\) 0 0
\(367\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.00000 1.00000
\(373\) −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
\(377\) 0 0
\(378\) 0 0
\(379\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(380\) 0 0
\(381\) 1.30902 0.951057i 1.30902 0.951057i
\(382\) 0 0
\(383\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(388\) −0.500000 1.53884i −0.500000 1.53884i
\(389\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(398\) 0 0
\(399\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(400\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(401\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(402\) 0 0
\(403\) 1.30902 0.951057i 1.30902 0.951057i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(418\) 0 0
\(419\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(420\) 0 0
\(421\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(432\) 0.309017 0.951057i 0.309017 0.951057i
\(433\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.190983 0.587785i 0.190983 0.587785i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(440\) 0 0
\(441\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(442\) 0 0
\(443\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(444\) 0.190983 0.587785i 0.190983 0.587785i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(449\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.190983 0.587785i 0.190983 0.587785i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(462\) 0 0
\(463\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(468\) −0.500000 1.53884i −0.500000 1.53884i
\(469\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(470\) 0 0
\(471\) −1.61803 −1.61803
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(480\) 0 0
\(481\) −0.309017 0.951057i −0.309017 0.951057i
\(482\) 0 0
\(483\) 0 0
\(484\) 0.309017 0.951057i 0.309017 0.951057i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(488\) 0 0
\(489\) −0.500000 0.363271i −0.500000 0.363271i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(497\) 0 0
\(498\) 0 0
\(499\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.30902 0.951057i −1.30902 0.951057i
\(508\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(509\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(510\) 0 0
\(511\) 0.118034 + 0.363271i 0.118034 + 0.363271i
\(512\) 0 0
\(513\) 1.30902 0.951057i 1.30902 0.951057i
\(514\) 0 0
\(515\) 0 0
\(516\) −0.500000 0.363271i −0.500000 0.363271i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(524\) 0 0
\(525\) 0.190983 0.587785i 0.190983 0.587785i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.309017 0.951057i 0.309017 0.951057i
\(530\) 0 0
\(531\) 0 0
\(532\) −1.00000 −1.00000
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(542\) 0 0
\(543\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(548\) 0 0
\(549\) 1.30902 0.951057i 1.30902 0.951057i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.00000 −1.00000
\(554\) 0 0
\(555\) 0 0
\(556\) 0.190983 0.587785i 0.190983 0.587785i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1.00000 −1.00000
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(568\) 0 0
\(569\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(570\) 0 0
\(571\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(577\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(578\) 0 0
\(579\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(588\) −0.618034 −0.618034
\(589\) −1.61803 −1.61803
\(590\) 0 0
\(591\) 0 0
\(592\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(593\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(598\) 0 0
\(599\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(600\) 0 0
\(601\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(602\) 0 0
\(603\) 1.30902 0.951057i 1.30902 0.951057i
\(604\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(605\) 0 0
\(606\) 0 0
\(607\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(618\) 0 0
\(619\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(625\) 1.00000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 1.30902 0.951057i 1.30902 0.951057i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(632\) 0 0
\(633\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(642\) 0 0
\(643\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.618034 0.618034
\(652\) 0.618034 0.618034
\(653\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.618034 0.618034
\(658\) 0 0
\(659\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(660\) 0 0
\(661\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −0.500000 1.53884i −0.500000 1.53884i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(674\) 0 0
\(675\) −0.809017 0.587785i −0.809017 0.587785i
\(676\) 1.61803 1.61803
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −0.309017 0.951057i −0.309017 0.951057i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(685\) 0 0
\(686\) 0 0
\(687\) −1.61803 −1.61803
\(688\) 0.618034 0.618034
\(689\) 0 0
\(690\) 0 0
\(691\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(701\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(702\) 0 0
\(703\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(710\) 0 0
\(711\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −0.309017 + 0.224514i −0.309017 + 0.224514i
\(722\) 0 0
\(723\) 0.618034 1.90211i 0.618034 1.90211i
\(724\) −0.500000 0.363271i −0.500000 0.363271i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(728\) 0 0
\(729\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(730\) 0 0
\(731\) 0 0
\(732\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(733\) −1.61803 1.17557i −1.61803 1.17557i −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 0
\(739\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) 0 0
\(741\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.190983 0.587785i 0.190983 0.587785i
\(757\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(762\) 0 0
\(763\) 0.118034 0.363271i 0.118034 0.363271i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.809017 0.587785i −0.809017 0.587785i
\(769\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.500000 1.53884i −0.500000 1.53884i
\(773\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(774\) 0 0
\(775\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(776\) 0 0
\(777\) 0.118034 0.363271i 0.118034 0.363271i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.500000 0.363271i 0.500000 0.363271i
\(785\) 0 0
\(786\) 0 0
\(787\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(794\) 0 0
\(795\) 0 0
\(796\) 0.190983 0.587785i 0.190983 0.587785i
\(797\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(810\) 0 0
\(811\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(812\) 0 0
\(813\) −0.500000 0.363271i −0.500000 0.363271i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(818\) 0 0
\(819\) −0.309017 0.951057i −0.309017 0.951057i
\(820\) 0 0
\(821\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(822\) 0 0
\(823\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(828\) 0 0
\(829\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(830\) 0 0
\(831\) 2.00000 2.00000
\(832\) −1.61803 −1.61803
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.309017 0.951057i 0.309017 0.951057i
\(838\) 0 0
\(839\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(840\) 0 0
\(841\) −0.809017 0.587785i −0.809017 0.587785i
\(842\) 0 0
\(843\) 0 0
\(844\) −0.500000 0.363271i −0.500000 0.363271i
\(845\) 0 0
\(846\) 0 0
\(847\) 0.190983 0.587785i 0.190983 0.587785i
\(848\) 0 0
\(849\) 0.190983 0.587785i 0.190983 0.587785i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(858\) 0 0
\(859\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000 1.00000
\(868\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(869\) 0 0
\(870\) 0 0
\(871\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(872\) 0 0
\(873\) −1.61803 −1.61803
\(874\) 0 0
\(875\) 0 0
\(876\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(877\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(882\) 0 0
\(883\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(888\) 0 0
\(889\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(890\) 0 0
\(891\) 0 0
\(892\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.00000 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −0.309017 0.224514i −0.309017 0.224514i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(912\) −0.500000 1.53884i −0.500000 1.53884i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.30902 0.951057i 1.30902 0.951057i
\(917\) 0 0
\(918\) 0 0
\(919\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(920\) 0 0
\(921\) 1.30902 0.951057i 1.30902 0.951057i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.618034 0.618034
\(926\) 0 0
\(927\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.00000 1.00000
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(938\) 0 0
\(939\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(940\) 0 0
\(941\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(948\) −0.500000 1.53884i −0.500000 1.53884i
\(949\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(962\) 0 0
\(963\) 0 0
\(964\) 0.618034 + 1.90211i 0.618034 + 1.90211i
\(965\) 0 0
\(966\) 0 0
\(967\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(972\) −0.809017 0.587785i −0.809017 0.587785i
\(973\) 0.118034 0.363271i 0.118034 0.363271i
\(974\) 0 0
\(975\) 1.30902 0.951057i 1.30902 0.951057i
\(976\) −0.500000 1.53884i −0.500000 1.53884i
\(977\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.500000 0.363271i −0.500000 0.363271i
\(982\) 0 0
\(983\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −2.11803 1.53884i −2.11803 1.53884i
\(989\) 0 0
\(990\) 0 0
\(991\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(992\) 0 0
\(993\) 2.00000 2.00000
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(998\) 0 0
\(999\) −0.500000 0.363271i −0.500000 0.363271i
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 93.1.l.a.8.1 4
3.2 odd 2 CM 93.1.l.a.8.1 4
4.3 odd 2 1488.1.br.a.1217.1 4
5.2 odd 4 2325.1.bq.a.1124.2 8
5.3 odd 4 2325.1.bq.a.1124.1 8
5.4 even 2 2325.1.ca.a.101.1 4
9.2 odd 6 2511.1.bu.a.2240.1 8
9.4 even 3 2511.1.bu.a.1403.1 8
9.5 odd 6 2511.1.bu.a.1403.1 8
9.7 even 3 2511.1.bu.a.2240.1 8
12.11 even 2 1488.1.br.a.1217.1 4
15.2 even 4 2325.1.bq.a.1124.2 8
15.8 even 4 2325.1.bq.a.1124.1 8
15.14 odd 2 2325.1.ca.a.101.1 4
31.2 even 5 2883.1.b.b.962.2 2
31.3 odd 30 2883.1.o.a.2738.1 8
31.4 even 5 inner 93.1.l.a.35.1 yes 4
31.5 even 3 2883.1.o.d.1409.1 8
31.6 odd 6 2883.1.o.b.2768.1 8
31.7 even 15 2883.1.o.d.1508.1 8
31.8 even 5 2883.1.l.a.1349.1 4
31.9 even 15 2883.1.o.c.338.1 8
31.10 even 15 2883.1.h.a.521.1 4
31.11 odd 30 2883.1.o.b.1805.1 8
31.12 odd 30 2883.1.h.b.1400.1 4
31.13 odd 30 2883.1.o.a.2654.1 8
31.14 even 15 2883.1.o.c.1196.1 8
31.15 odd 10 2883.1.l.c.374.1 4
31.16 even 5 2883.1.l.a.374.1 4
31.17 odd 30 2883.1.o.a.1196.1 8
31.18 even 15 2883.1.o.c.2654.1 8
31.19 even 15 2883.1.h.a.1400.1 4
31.20 even 15 2883.1.o.d.1805.1 8
31.21 odd 30 2883.1.h.b.521.1 4
31.22 odd 30 2883.1.o.a.338.1 8
31.23 odd 10 2883.1.l.c.1349.1 4
31.24 odd 30 2883.1.o.b.1508.1 8
31.25 even 3 2883.1.o.d.2768.1 8
31.26 odd 6 2883.1.o.b.1409.1 8
31.27 odd 10 2883.1.l.b.2453.1 4
31.28 even 15 2883.1.o.c.2738.1 8
31.29 odd 10 2883.1.b.a.962.2 2
31.30 odd 2 2883.1.l.b.1589.1 4
93.2 odd 10 2883.1.b.b.962.2 2
93.5 odd 6 2883.1.o.d.1409.1 8
93.8 odd 10 2883.1.l.a.1349.1 4
93.11 even 30 2883.1.o.b.1805.1 8
93.14 odd 30 2883.1.o.c.1196.1 8
93.17 even 30 2883.1.o.a.1196.1 8
93.20 odd 30 2883.1.o.d.1805.1 8
93.23 even 10 2883.1.l.c.1349.1 4
93.26 even 6 2883.1.o.b.1409.1 8
93.29 even 10 2883.1.b.a.962.2 2
93.35 odd 10 inner 93.1.l.a.35.1 yes 4
93.38 odd 30 2883.1.o.d.1508.1 8
93.41 odd 30 2883.1.h.a.521.1 4
93.44 even 30 2883.1.o.a.2654.1 8
93.47 odd 10 2883.1.l.a.374.1 4
93.50 odd 30 2883.1.h.a.1400.1 4
93.53 even 30 2883.1.o.a.338.1 8
93.56 odd 6 2883.1.o.d.2768.1 8
93.59 odd 30 2883.1.o.c.2738.1 8
93.65 even 30 2883.1.o.a.2738.1 8
93.68 even 6 2883.1.o.b.2768.1 8
93.71 odd 30 2883.1.o.c.338.1 8
93.74 even 30 2883.1.h.b.1400.1 4
93.77 even 10 2883.1.l.c.374.1 4
93.80 odd 30 2883.1.o.c.2654.1 8
93.83 even 30 2883.1.h.b.521.1 4
93.86 even 30 2883.1.o.b.1508.1 8
93.89 even 10 2883.1.l.b.2453.1 4
93.92 even 2 2883.1.l.b.1589.1 4
124.35 odd 10 1488.1.br.a.593.1 4
155.4 even 10 2325.1.ca.a.1151.1 4
155.97 odd 20 2325.1.bq.a.2174.1 8
155.128 odd 20 2325.1.bq.a.2174.2 8
279.4 even 15 2511.1.bu.a.593.1 8
279.97 even 15 2511.1.bu.a.1430.1 8
279.128 odd 30 2511.1.bu.a.1430.1 8
279.221 odd 30 2511.1.bu.a.593.1 8
372.35 even 10 1488.1.br.a.593.1 4
465.128 even 20 2325.1.bq.a.2174.2 8
465.314 odd 10 2325.1.ca.a.1151.1 4
465.407 even 20 2325.1.bq.a.2174.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
93.1.l.a.8.1 4 1.1 even 1 trivial
93.1.l.a.8.1 4 3.2 odd 2 CM
93.1.l.a.35.1 yes 4 31.4 even 5 inner
93.1.l.a.35.1 yes 4 93.35 odd 10 inner
1488.1.br.a.593.1 4 124.35 odd 10
1488.1.br.a.593.1 4 372.35 even 10
1488.1.br.a.1217.1 4 4.3 odd 2
1488.1.br.a.1217.1 4 12.11 even 2
2325.1.bq.a.1124.1 8 5.3 odd 4
2325.1.bq.a.1124.1 8 15.8 even 4
2325.1.bq.a.1124.2 8 5.2 odd 4
2325.1.bq.a.1124.2 8 15.2 even 4
2325.1.bq.a.2174.1 8 155.97 odd 20
2325.1.bq.a.2174.1 8 465.407 even 20
2325.1.bq.a.2174.2 8 155.128 odd 20
2325.1.bq.a.2174.2 8 465.128 even 20
2325.1.ca.a.101.1 4 5.4 even 2
2325.1.ca.a.101.1 4 15.14 odd 2
2325.1.ca.a.1151.1 4 155.4 even 10
2325.1.ca.a.1151.1 4 465.314 odd 10
2511.1.bu.a.593.1 8 279.4 even 15
2511.1.bu.a.593.1 8 279.221 odd 30
2511.1.bu.a.1403.1 8 9.4 even 3
2511.1.bu.a.1403.1 8 9.5 odd 6
2511.1.bu.a.1430.1 8 279.97 even 15
2511.1.bu.a.1430.1 8 279.128 odd 30
2511.1.bu.a.2240.1 8 9.2 odd 6
2511.1.bu.a.2240.1 8 9.7 even 3
2883.1.b.a.962.2 2 31.29 odd 10
2883.1.b.a.962.2 2 93.29 even 10
2883.1.b.b.962.2 2 31.2 even 5
2883.1.b.b.962.2 2 93.2 odd 10
2883.1.h.a.521.1 4 31.10 even 15
2883.1.h.a.521.1 4 93.41 odd 30
2883.1.h.a.1400.1 4 31.19 even 15
2883.1.h.a.1400.1 4 93.50 odd 30
2883.1.h.b.521.1 4 31.21 odd 30
2883.1.h.b.521.1 4 93.83 even 30
2883.1.h.b.1400.1 4 31.12 odd 30
2883.1.h.b.1400.1 4 93.74 even 30
2883.1.l.a.374.1 4 31.16 even 5
2883.1.l.a.374.1 4 93.47 odd 10
2883.1.l.a.1349.1 4 31.8 even 5
2883.1.l.a.1349.1 4 93.8 odd 10
2883.1.l.b.1589.1 4 31.30 odd 2
2883.1.l.b.1589.1 4 93.92 even 2
2883.1.l.b.2453.1 4 31.27 odd 10
2883.1.l.b.2453.1 4 93.89 even 10
2883.1.l.c.374.1 4 31.15 odd 10
2883.1.l.c.374.1 4 93.77 even 10
2883.1.l.c.1349.1 4 31.23 odd 10
2883.1.l.c.1349.1 4 93.23 even 10
2883.1.o.a.338.1 8 31.22 odd 30
2883.1.o.a.338.1 8 93.53 even 30
2883.1.o.a.1196.1 8 31.17 odd 30
2883.1.o.a.1196.1 8 93.17 even 30
2883.1.o.a.2654.1 8 31.13 odd 30
2883.1.o.a.2654.1 8 93.44 even 30
2883.1.o.a.2738.1 8 31.3 odd 30
2883.1.o.a.2738.1 8 93.65 even 30
2883.1.o.b.1409.1 8 31.26 odd 6
2883.1.o.b.1409.1 8 93.26 even 6
2883.1.o.b.1508.1 8 31.24 odd 30
2883.1.o.b.1508.1 8 93.86 even 30
2883.1.o.b.1805.1 8 31.11 odd 30
2883.1.o.b.1805.1 8 93.11 even 30
2883.1.o.b.2768.1 8 31.6 odd 6
2883.1.o.b.2768.1 8 93.68 even 6
2883.1.o.c.338.1 8 31.9 even 15
2883.1.o.c.338.1 8 93.71 odd 30
2883.1.o.c.1196.1 8 31.14 even 15
2883.1.o.c.1196.1 8 93.14 odd 30
2883.1.o.c.2654.1 8 31.18 even 15
2883.1.o.c.2654.1 8 93.80 odd 30
2883.1.o.c.2738.1 8 31.28 even 15
2883.1.o.c.2738.1 8 93.59 odd 30
2883.1.o.d.1409.1 8 31.5 even 3
2883.1.o.d.1409.1 8 93.5 odd 6
2883.1.o.d.1508.1 8 31.7 even 15
2883.1.o.d.1508.1 8 93.38 odd 30
2883.1.o.d.1805.1 8 31.20 even 15
2883.1.o.d.1805.1 8 93.20 odd 30
2883.1.o.d.2768.1 8 31.25 even 3
2883.1.o.d.2768.1 8 93.56 odd 6