Properties

Label 88.1.l.a.3.1
Level $88$
Weight $1$
Character 88.3
Analytic conductor $0.044$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [88,1,Mod(3,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 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("88.3");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 88.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: \(0.0439177211117\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.937024.1

Embedding invariants

Embedding label 3.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 88.3
Dual form 88.1.l.a.59.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.809017 + 0.587785i) q^{2} +(-0.500000 - 1.53884i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.30902 + 0.951057i) q^{6} +(0.309017 + 0.951057i) q^{8} +(-1.30902 + 0.951057i) q^{9} +O(q^{10})\) \(q+(-0.809017 + 0.587785i) q^{2} +(-0.500000 - 1.53884i) q^{3} +(0.309017 - 0.951057i) q^{4} +(1.30902 + 0.951057i) q^{6} +(0.309017 + 0.951057i) q^{8} +(-1.30902 + 0.951057i) q^{9} +(0.309017 + 0.951057i) q^{11} -1.61803 q^{12} +(-0.809017 - 0.587785i) q^{16} +(-0.500000 - 0.363271i) q^{17} +(0.500000 - 1.53884i) q^{18} +(0.190983 + 0.587785i) q^{19} +(-0.809017 - 0.587785i) q^{22} +(1.30902 - 0.951057i) q^{24} +(0.309017 + 0.951057i) q^{25} +(0.809017 + 0.587785i) q^{27} +1.00000 q^{32} +(1.30902 - 0.951057i) q^{33} +0.618034 q^{34} +(0.500000 + 1.53884i) q^{36} +(-0.500000 - 0.363271i) q^{38} +(-0.500000 - 1.53884i) q^{41} -1.61803 q^{43} +1.00000 q^{44} +(-0.500000 + 1.53884i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-0.809017 - 0.587785i) q^{50} +(-0.309017 + 0.951057i) q^{51} -1.00000 q^{54} +(0.809017 - 0.587785i) q^{57} +(0.190983 - 0.587785i) q^{59} +(-0.809017 + 0.587785i) q^{64} +(-0.500000 + 1.53884i) q^{66} +0.618034 q^{67} +(-0.500000 + 0.363271i) q^{68} +(-1.30902 - 0.951057i) q^{72} +(-0.500000 + 1.53884i) q^{73} +(1.30902 - 0.951057i) q^{75} +0.618034 q^{76} +(1.30902 + 0.951057i) q^{82} +(1.30902 + 0.951057i) q^{83} +(1.30902 - 0.951057i) q^{86} +(-0.809017 + 0.587785i) q^{88} -1.61803 q^{89} +(-0.500000 - 1.53884i) q^{96} +(1.30902 - 0.951057i) q^{97} +1.00000 q^{98} +(-1.30902 - 0.951057i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 2 q^{3} - q^{4} + 3 q^{6} - q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 2 q^{3} - q^{4} + 3 q^{6} - q^{8} - 3 q^{9} - q^{11} - 2 q^{12} - q^{16} - 2 q^{17} + 2 q^{18} + 3 q^{19} - q^{22} + 3 q^{24} - q^{25} + q^{27} + 4 q^{32} + 3 q^{33} - 2 q^{34} + 2 q^{36} - 2 q^{38} - 2 q^{41} - 2 q^{43} + 4 q^{44} - 2 q^{48} - q^{49} - q^{50} + q^{51} - 4 q^{54} + q^{57} + 3 q^{59} - q^{64} - 2 q^{66} - 2 q^{67} - 2 q^{68} - 3 q^{72} - 2 q^{73} + 3 q^{75} - 2 q^{76} + 3 q^{82} + 3 q^{83} + 3 q^{86} - q^{88} - 2 q^{89} - 2 q^{96} + 3 q^{97} + 4 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).

\(n\) \(23\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{4}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(3\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(4\) 0.309017 0.951057i 0.309017 0.951057i
\(5\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(6\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(7\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(8\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(9\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(10\) 0 0
\(11\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(12\) −1.61803 −1.61803
\(13\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.809017 0.587785i −0.809017 0.587785i
\(17\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(18\) 0.500000 1.53884i 0.500000 1.53884i
\(19\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.809017 0.587785i −0.809017 0.587785i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.30902 0.951057i 1.30902 0.951057i
\(25\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(26\) 0 0
\(27\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(28\) 0 0
\(29\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(30\) 0 0
\(31\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(32\) 1.00000 1.00000
\(33\) 1.30902 0.951057i 1.30902 0.951057i
\(34\) 0.618034 0.618034
\(35\) 0 0
\(36\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(37\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(38\) −0.500000 0.363271i −0.500000 0.363271i
\(39\) 0 0
\(40\) 0 0
\(41\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(42\) 0 0
\(43\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(44\) 1.00000 1.00000
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(48\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(49\) −0.809017 0.587785i −0.809017 0.587785i
\(50\) −0.809017 0.587785i −0.809017 0.587785i
\(51\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(52\) 0 0
\(53\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) 0.809017 0.587785i 0.809017 0.587785i
\(58\) 0 0
\(59\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(60\) 0 0
\(61\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(65\) 0 0
\(66\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(67\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(68\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(72\) −1.30902 0.951057i −1.30902 0.951057i
\(73\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(74\) 0 0
\(75\) 1.30902 0.951057i 1.30902 0.951057i
\(76\) 0.618034 0.618034
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(83\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.30902 0.951057i 1.30902 0.951057i
\(87\) 0 0
\(88\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(89\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −0.500000 1.53884i −0.500000 1.53884i
\(97\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(98\) 1.00000 1.00000
\(99\) −1.30902 0.951057i −1.30902 0.951057i
\(100\) 1.00000 1.00000
\(101\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(102\) −0.309017 0.951057i −0.309017 0.951057i
\(103\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(108\) 0.809017 0.587785i 0.809017 0.587785i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(114\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(122\) 0 0
\(123\) −2.11803 + 1.53884i −2.11803 + 1.53884i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(128\) 0.309017 0.951057i 0.309017 0.951057i
\(129\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(130\) 0 0
\(131\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(132\) −0.500000 1.53884i −0.500000 1.53884i
\(133\) 0 0
\(134\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(135\) 0 0
\(136\) 0.190983 0.587785i 0.190983 0.587785i
\(137\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(138\) 0 0
\(139\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.61803 1.61803
\(145\) 0 0
\(146\) −0.500000 1.53884i −0.500000 1.53884i
\(147\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(148\) 0 0
\(149\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(150\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(151\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(152\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(153\) 1.00000 1.00000
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(158\) 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 \(0.400000\pi\)
\(164\) −1.61803 −1.61803
\(165\) 0 0
\(166\) −1.61803 −1.61803
\(167\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(168\) 0 0
\(169\) 0.309017 0.951057i 0.309017 0.951057i
\(170\) 0 0
\(171\) −0.809017 0.587785i −0.809017 0.587785i
\(172\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(173\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.309017 0.951057i 0.309017 0.951057i
\(177\) −1.00000 −1.00000
\(178\) 1.30902 0.951057i 1.30902 0.951057i
\(179\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(180\) 0 0
\(181\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.190983 0.587785i 0.190983 0.587785i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(192\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(193\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(194\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(195\) 0 0
\(196\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 1.61803 1.61803
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(201\) −0.309017 0.951057i −0.309017 0.951057i
\(202\) 0 0
\(203\) 0 0
\(204\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(210\) 0 0
\(211\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(215\) 0 0
\(216\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(217\) 0 0
\(218\) 0 0
\(219\) 2.61803 2.61803
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(224\) 0 0
\(225\) −1.30902 0.951057i −1.30902 0.951057i
\(226\) −0.500000 0.363271i −0.500000 0.363271i
\(227\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(228\) −0.309017 0.951057i −0.309017 0.951057i
\(229\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.500000 0.363271i −0.500000 0.363271i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(240\) 0 0
\(241\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(242\) 0.309017 0.951057i 0.309017 0.951057i
\(243\) 1.00000 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0.809017 2.48990i 0.809017 2.48990i
\(247\) 0 0
\(248\) 0 0
\(249\) 0.809017 2.48990i 0.809017 2.48990i
\(250\) 0 0
\(251\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(257\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(258\) −2.11803 1.53884i −2.11803 1.53884i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(265\) 0 0
\(266\) 0 0
\(267\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(268\) 0.190983 0.587785i 0.190983 0.587785i
\(269\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(270\) 0 0
\(271\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(272\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(273\) 0 0
\(274\) −1.61803 −1.61803
\(275\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(276\) 0 0
\(277\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(278\) 0.618034 + 1.90211i 0.618034 + 1.90211i
\(279\) 0 0
\(280\) 0 0
\(281\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(282\) 0 0
\(283\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(289\) −0.190983 0.587785i −0.190983 0.587785i
\(290\) 0 0
\(291\) −2.11803 1.53884i −2.11803 1.53884i
\(292\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(293\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(294\) −0.500000 1.53884i −0.500000 1.53884i
\(295\) 0 0
\(296\) 0 0
\(297\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(298\) 0 0
\(299\) 0 0
\(300\) −0.500000 1.53884i −0.500000 1.53884i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0.190983 0.587785i 0.190983 0.587785i
\(305\) 0 0
\(306\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(307\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(312\) 0 0
\(313\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −2.11803 + 1.53884i −2.11803 + 1.53884i
\(322\) 0 0
\(323\) 0.118034 0.363271i 0.118034 0.363271i
\(324\) 0 0
\(325\) 0 0
\(326\) 0.190983 0.587785i 0.190983 0.587785i
\(327\) 0 0
\(328\) 1.30902 0.951057i 1.30902 0.951057i
\(329\) 0 0
\(330\) 0 0
\(331\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(332\) 1.30902 0.951057i 1.30902 0.951057i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(338\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(339\) 0.809017 0.587785i 0.809017 0.587785i
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000 1.00000
\(343\) 0 0
\(344\) −0.500000 1.53884i −0.500000 1.53884i
\(345\) 0 0
\(346\) 0 0
\(347\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(353\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(354\) 0.809017 0.587785i 0.809017 0.587785i
\(355\) 0 0
\(356\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(357\) 0 0
\(358\) −0.500000 0.363271i −0.500000 0.363271i
\(359\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(360\) 0 0
\(361\) 0.500000 0.363271i 0.500000 0.363271i
\(362\) 0 0
\(363\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(368\) 0 0
\(369\) 2.11803 + 1.53884i 2.11803 + 1.53884i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(375\) 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
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(384\) −1.61803 −1.61803
\(385\) 0 0
\(386\) 2.00000 2.00000
\(387\) 2.11803 1.53884i 2.11803 1.53884i
\(388\) −0.500000 1.53884i −0.500000 1.53884i
\(389\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.309017 0.951057i 0.309017 0.951057i
\(393\) −0.309017 0.951057i −0.309017 0.951057i
\(394\) 0 0
\(395\) 0 0
\(396\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.309017 0.951057i 0.309017 0.951057i
\(401\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(402\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −1.00000 −1.00000
\(409\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(410\) 0 0
\(411\) 0.809017 2.48990i 0.809017 2.48990i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.23607 −3.23607
\(418\) 0.190983 0.587785i 0.190983 0.587785i
\(419\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(420\) 0 0
\(421\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(423\) 0 0
\(424\) 0 0
\(425\) 0.190983 0.587785i 0.190983 0.587785i
\(426\) 0 0
\(427\) 0 0
\(428\) −1.61803 −1.61803
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(432\) −0.309017 0.951057i −0.309017 0.951057i
\(433\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −2.11803 + 1.53884i −2.11803 + 1.53884i
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.61803 1.61803
\(442\) 0 0
\(443\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(450\) 1.61803 1.61803
\(451\) 1.30902 0.951057i 1.30902 0.951057i
\(452\) 0.618034 0.618034
\(453\) 0 0
\(454\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(455\) 0 0
\(456\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(457\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(458\) 0 0
\(459\) −0.190983 0.587785i −0.190983 0.587785i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(467\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0.618034 0.618034
\(473\) −0.500000 1.53884i −0.500000 1.53884i
\(474\) 0 0
\(475\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(483\) 0 0
\(484\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(485\) 0 0
\(486\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(487\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(488\) 0 0
\(489\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(490\) 0 0
\(491\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(492\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(499\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.618034 1.90211i 0.618034 1.90211i
\(503\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.61803 −1.61803
\(508\) 0 0
\(509\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.809017 0.587785i −0.809017 0.587785i
\(513\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(514\) −0.500000 1.53884i −0.500000 1.53884i
\(515\) 0 0
\(516\) 2.61803 2.61803
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(522\) 0 0
\(523\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(524\) 0.190983 0.587785i 0.190983 0.587785i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −1.61803 −1.61803
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(532\) 0 0
\(533\) 0 0
\(534\) −2.11803 1.53884i −2.11803 1.53884i
\(535\) 0 0
\(536\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(537\) 0.809017 0.587785i 0.809017 0.587785i
\(538\) 0 0
\(539\) 0.309017 0.951057i 0.309017 0.951057i
\(540\) 0 0
\(541\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.500000 0.363271i −0.500000 0.363271i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(548\) 1.30902 0.951057i 1.30902 0.951057i
\(549\) 0 0
\(550\) 0.309017 0.951057i 0.309017 0.951057i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.61803 1.17557i −1.61803 1.17557i
\(557\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −1.00000 −1.00000
\(562\) −1.61803 −1.61803
\(563\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.61803 1.17557i −1.61803 1.17557i
\(567\) 0 0
\(568\) 0 0
\(569\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(570\) 0 0
\(571\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.500000 1.53884i 0.500000 1.53884i
\(577\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(578\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(579\) −1.00000 + 3.07768i −1.00000 + 3.07768i
\(580\) 0 0
\(581\) 0 0
\(582\) 2.61803 2.61803
\(583\) 0 0
\(584\) −1.61803 −1.61803
\(585\) 0 0
\(586\) 0 0
\(587\) 0.190983 0.587785i 0.190983 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
1.00000 \(0\)
\(588\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(594\) −0.309017 0.951057i −0.309017 0.951057i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(600\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(601\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(602\) 0 0
\(603\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(608\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0.309017 0.951057i 0.309017 0.951057i
\(613\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(614\) 1.30902 0.951057i 1.30902 0.951057i
\(615\) 0 0
\(616\) 0 0
\(617\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(618\) 0 0
\(619\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(626\) 0.618034 0.618034
\(627\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(632\) 0 0
\(633\) −2.11803 1.53884i −2.11803 1.53884i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(642\) 0.809017 2.48990i 0.809017 2.48990i
\(643\) 1.30902 + 0.951057i 1.30902 + 0.951057i 1.00000 \(0\)
0.309017 + 0.951057i \(0.400000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.118034 + 0.363271i 0.118034 + 0.363271i
\(647\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(648\) 0 0
\(649\) 0.618034 0.618034
\(650\) 0 0
\(651\) 0 0
\(652\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(653\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(657\) −0.809017 2.48990i −0.809017 2.48990i
\(658\) 0 0
\(659\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 1.30902 0.951057i 1.30902 0.951057i
\(663\) 0 0
\(664\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(674\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(675\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(676\) −0.809017 0.587785i −0.809017 0.587785i
\(677\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(678\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(679\) 0 0
\(680\) 0 0
\(681\) −1.00000 −1.00000
\(682\) 0 0
\(683\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(684\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(689\) 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 0
\(694\) 0.618034 0.618034
\(695\) 0 0
\(696\) 0 0
\(697\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(698\) 0 0
\(699\) −2.11803 1.53884i −2.11803 1.53884i
\(700\) 0 0
\(701\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.809017 0.587785i −0.809017 0.587785i
\(705\) 0 0
\(706\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(707\) 0 0
\(708\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(709\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.500000 1.53884i −0.500000 1.53884i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.618034 0.618034
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.190983 + 0.587785i −0.190983 + 0.587785i
\(723\) −0.309017 0.951057i −0.309017 0.951057i
\(724\) 0 0
\(725\) 0 0
\(726\) −1.61803 −1.61803
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −0.500000 1.53884i −0.500000 1.53884i
\(730\) 0 0
\(731\) 0.809017 + 0.587785i 0.809017 + 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
\(737\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(738\) −2.61803 −2.61803
\(739\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −2.61803 −2.61803
\(748\) −0.500000 0.363271i −0.500000 0.363271i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(752\) 0 0
\(753\) 2.61803 + 1.90211i 2.61803 + 1.90211i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(758\) −1.61803 −1.61803
\(759\) 0 0
\(760\) 0 0
\(761\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.30902 0.951057i 1.30902 0.951057i
\(769\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(770\) 0 0
\(771\) 2.61803 2.61803
\(772\) −1.61803 + 1.17557i −1.61803 + 1.17557i
\(773\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(774\) −0.809017 + 2.48990i −0.809017 + 2.48990i
\(775\) 0 0
\(776\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(777\) 0 0
\(778\) 0 0
\(779\) 0.809017 0.587785i 0.809017 0.587785i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(785\) 0 0
\(786\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(787\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.500000 1.53884i 0.500000 1.53884i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(801\) 2.11803 1.53884i 2.11803 1.53884i
\(802\) −1.61803 −1.61803
\(803\) −1.61803 −1.61803
\(804\) −1.00000 −1.00000
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(810\) 0 0
\(811\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0.809017 0.587785i 0.809017 0.587785i
\(817\) −0.309017 0.951057i −0.309017 0.951057i
\(818\) 0.618034 1.90211i 0.618034 1.90211i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(822\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(823\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(824\) 0 0
\(825\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(826\) 0 0
\(827\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(828\) 0 0
\(829\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(834\) 2.61803 1.90211i 2.61803 1.90211i
\(835\) 0 0
\(836\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(837\) 0 0
\(838\) 1.30902 0.951057i 1.30902 0.951057i
\(839\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(840\) 0 0
\(841\) −0.809017 0.587785i −0.809017 0.587785i
\(842\) 0 0
\(843\) 0.809017 2.48990i 0.809017 2.48990i
\(844\) −0.500000 1.53884i −0.500000 1.53884i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2.61803 1.90211i 2.61803 1.90211i
\(850\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.30902 0.951057i 1.30902 0.951057i
\(857\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(858\) 0 0
\(859\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(864\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(865\) 0 0
\(866\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(867\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.809017 + 2.48990i −0.809017 + 2.48990i
\(874\) 0 0
\(875\) 0 0
\(876\) 0.809017 2.48990i 0.809017 2.48990i
\(877\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(882\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(883\) −0.500000 1.53884i −0.500000 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(887\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0.190983 0.587785i 0.190983 0.587785i
\(899\) 0 0
\(900\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(901\) 0 0
\(902\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(903\) 0 0
\(904\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(908\) −0.500000 0.363271i −0.500000 0.363271i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(912\) −1.00000 −1.00000
\(913\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(914\) −1.61803 −1.61803
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(919\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(920\) 0 0
\(921\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(930\) 0 0
\(931\) 0.190983 0.587785i 0.190983 0.587785i
\(932\) −0.500000 1.53884i −0.500000 1.53884i
\(933\) 0 0
\(934\) 2.00000 2.00000
\(935\) 0 0
\(936\) 0 0
\(937\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(938\) 0 0
\(939\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(940\) 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
\(944\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(945\) 0 0
\(946\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(947\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.190983 0.587785i 0.190983 0.587785i
\(951\) 0 0
\(952\) 0 0
\(953\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−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
\(961\) 0.309017 0.951057i 0.309017 0.951057i
\(962\) 0 0
\(963\) 2.11803 + 1.53884i 2.11803 + 1.53884i
\(964\) 0.190983 0.587785i 0.190983 0.587785i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −0.809017 0.587785i −0.809017 0.587785i
\(969\) −0.618034 −0.618034
\(970\) 0 0
\(971\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(972\) 0.309017 0.951057i 0.309017 0.951057i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(978\) −1.00000 −1.00000
\(979\) −0.500000 1.53884i −0.500000 1.53884i
\(980\) 0 0
\(981\) 0 0
\(982\) −0.500000 1.53884i −0.500000 1.53884i
\(983\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(984\) −2.11803 1.53884i −2.11803 1.53884i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0.809017 + 2.48990i 0.809017 + 2.48990i
\(994\) 0 0
\(995\) 0 0
\(996\) −2.11803 1.53884i −2.11803 1.53884i
\(997\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(998\) −0.500000 1.53884i −0.500000 1.53884i
\(999\) 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 88.1.l.a.3.1 4
3.2 odd 2 792.1.bu.a.91.1 4
4.3 odd 2 352.1.t.a.47.1 4
5.2 odd 4 2200.1.dd.a.1499.1 8
5.3 odd 4 2200.1.dd.a.1499.2 8
5.4 even 2 2200.1.cl.a.1851.1 4
8.3 odd 2 CM 88.1.l.a.3.1 4
8.5 even 2 352.1.t.a.47.1 4
11.2 odd 10 968.1.f.a.243.1 2
11.3 even 5 968.1.l.a.27.1 4
11.4 even 5 inner 88.1.l.a.59.1 yes 4
11.5 even 5 968.1.l.a.251.1 4
11.6 odd 10 968.1.l.c.251.1 4
11.7 odd 10 968.1.l.b.323.1 4
11.8 odd 10 968.1.l.c.27.1 4
11.9 even 5 968.1.f.b.243.1 2
11.10 odd 2 968.1.l.b.3.1 4
12.11 even 2 3168.1.ck.a.2863.1 4
16.3 odd 4 2816.1.v.c.1279.1 8
16.5 even 4 2816.1.v.c.1279.1 8
16.11 odd 4 2816.1.v.c.1279.2 8
16.13 even 4 2816.1.v.c.1279.2 8
24.5 odd 2 3168.1.ck.a.2863.1 4
24.11 even 2 792.1.bu.a.91.1 4
33.26 odd 10 792.1.bu.a.235.1 4
40.3 even 4 2200.1.dd.a.1499.2 8
40.19 odd 2 2200.1.cl.a.1851.1 4
40.27 even 4 2200.1.dd.a.1499.1 8
44.3 odd 10 3872.1.t.b.2447.1 4
44.7 even 10 3872.1.t.c.1775.1 4
44.15 odd 10 352.1.t.a.15.1 4
44.19 even 10 3872.1.t.a.2447.1 4
44.27 odd 10 3872.1.t.b.2671.1 4
44.31 odd 10 3872.1.f.a.3631.2 2
44.35 even 10 3872.1.f.b.3631.2 2
44.39 even 10 3872.1.t.a.2671.1 4
44.43 even 2 3872.1.t.c.1455.1 4
55.4 even 10 2200.1.cl.a.851.1 4
55.37 odd 20 2200.1.dd.a.499.2 8
55.48 odd 20 2200.1.dd.a.499.1 8
88.3 odd 10 968.1.l.a.27.1 4
88.5 even 10 3872.1.t.b.2671.1 4
88.13 odd 10 3872.1.f.b.3631.2 2
88.19 even 10 968.1.l.c.27.1 4
88.21 odd 2 3872.1.t.c.1455.1 4
88.27 odd 10 968.1.l.a.251.1 4
88.29 odd 10 3872.1.t.c.1775.1 4
88.35 even 10 968.1.f.a.243.1 2
88.37 even 10 352.1.t.a.15.1 4
88.43 even 2 968.1.l.b.3.1 4
88.51 even 10 968.1.l.b.323.1 4
88.53 even 10 3872.1.f.a.3631.2 2
88.59 odd 10 inner 88.1.l.a.59.1 yes 4
88.61 odd 10 3872.1.t.a.2671.1 4
88.69 even 10 3872.1.t.b.2447.1 4
88.75 odd 10 968.1.f.b.243.1 2
88.83 even 10 968.1.l.c.251.1 4
88.85 odd 10 3872.1.t.a.2447.1 4
132.59 even 10 3168.1.ck.a.1423.1 4
176.37 even 20 2816.1.v.c.2303.2 8
176.59 odd 20 2816.1.v.c.2303.1 8
176.125 even 20 2816.1.v.c.2303.1 8
176.147 odd 20 2816.1.v.c.2303.2 8
264.59 even 10 792.1.bu.a.235.1 4
264.125 odd 10 3168.1.ck.a.1423.1 4
440.59 odd 10 2200.1.cl.a.851.1 4
440.147 even 20 2200.1.dd.a.499.2 8
440.323 even 20 2200.1.dd.a.499.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.1.l.a.3.1 4 1.1 even 1 trivial
88.1.l.a.3.1 4 8.3 odd 2 CM
88.1.l.a.59.1 yes 4 11.4 even 5 inner
88.1.l.a.59.1 yes 4 88.59 odd 10 inner
352.1.t.a.15.1 4 44.15 odd 10
352.1.t.a.15.1 4 88.37 even 10
352.1.t.a.47.1 4 4.3 odd 2
352.1.t.a.47.1 4 8.5 even 2
792.1.bu.a.91.1 4 3.2 odd 2
792.1.bu.a.91.1 4 24.11 even 2
792.1.bu.a.235.1 4 33.26 odd 10
792.1.bu.a.235.1 4 264.59 even 10
968.1.f.a.243.1 2 11.2 odd 10
968.1.f.a.243.1 2 88.35 even 10
968.1.f.b.243.1 2 11.9 even 5
968.1.f.b.243.1 2 88.75 odd 10
968.1.l.a.27.1 4 11.3 even 5
968.1.l.a.27.1 4 88.3 odd 10
968.1.l.a.251.1 4 11.5 even 5
968.1.l.a.251.1 4 88.27 odd 10
968.1.l.b.3.1 4 11.10 odd 2
968.1.l.b.3.1 4 88.43 even 2
968.1.l.b.323.1 4 11.7 odd 10
968.1.l.b.323.1 4 88.51 even 10
968.1.l.c.27.1 4 11.8 odd 10
968.1.l.c.27.1 4 88.19 even 10
968.1.l.c.251.1 4 11.6 odd 10
968.1.l.c.251.1 4 88.83 even 10
2200.1.cl.a.851.1 4 55.4 even 10
2200.1.cl.a.851.1 4 440.59 odd 10
2200.1.cl.a.1851.1 4 5.4 even 2
2200.1.cl.a.1851.1 4 40.19 odd 2
2200.1.dd.a.499.1 8 55.48 odd 20
2200.1.dd.a.499.1 8 440.323 even 20
2200.1.dd.a.499.2 8 55.37 odd 20
2200.1.dd.a.499.2 8 440.147 even 20
2200.1.dd.a.1499.1 8 5.2 odd 4
2200.1.dd.a.1499.1 8 40.27 even 4
2200.1.dd.a.1499.2 8 5.3 odd 4
2200.1.dd.a.1499.2 8 40.3 even 4
2816.1.v.c.1279.1 8 16.3 odd 4
2816.1.v.c.1279.1 8 16.5 even 4
2816.1.v.c.1279.2 8 16.11 odd 4
2816.1.v.c.1279.2 8 16.13 even 4
2816.1.v.c.2303.1 8 176.59 odd 20
2816.1.v.c.2303.1 8 176.125 even 20
2816.1.v.c.2303.2 8 176.37 even 20
2816.1.v.c.2303.2 8 176.147 odd 20
3168.1.ck.a.1423.1 4 132.59 even 10
3168.1.ck.a.1423.1 4 264.125 odd 10
3168.1.ck.a.2863.1 4 12.11 even 2
3168.1.ck.a.2863.1 4 24.5 odd 2
3872.1.f.a.3631.2 2 44.31 odd 10
3872.1.f.a.3631.2 2 88.53 even 10
3872.1.f.b.3631.2 2 44.35 even 10
3872.1.f.b.3631.2 2 88.13 odd 10
3872.1.t.a.2447.1 4 44.19 even 10
3872.1.t.a.2447.1 4 88.85 odd 10
3872.1.t.a.2671.1 4 44.39 even 10
3872.1.t.a.2671.1 4 88.61 odd 10
3872.1.t.b.2447.1 4 44.3 odd 10
3872.1.t.b.2447.1 4 88.69 even 10
3872.1.t.b.2671.1 4 44.27 odd 10
3872.1.t.b.2671.1 4 88.5 even 10
3872.1.t.c.1455.1 4 44.43 even 2
3872.1.t.c.1455.1 4 88.21 odd 2
3872.1.t.c.1775.1 4 44.7 even 10
3872.1.t.c.1775.1 4 88.29 odd 10