Properties

Label 1859.1.k.c.1374.3
Level $1859$
Weight $1$
Character 1859.1374
Analytic conductor $0.928$
Analytic rank $0$
Dimension $8$
Projective image $D_{5}$
CM discriminant -143
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1859,1,Mod(1374,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1859.1374");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1859.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.927761858485\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 143)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.20449.1
Artin image: $D_5\times C_{12}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{60} - \cdots)\)

Embedding invariants

Embedding label 1374.3
Root \(-1.40126 + 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 1859.1374
Dual form 1859.1.k.c.1836.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.535233 + 0.309017i) q^{2} +(-0.309017 + 0.535233i) q^{3} +(-0.309017 - 0.535233i) q^{4} +(-0.330792 + 0.190983i) q^{6} +(-1.40126 + 0.809017i) q^{7} -1.00000i q^{8} +(0.309017 + 0.535233i) q^{9} +O(q^{10})\) \(q+(0.535233 + 0.309017i) q^{2} +(-0.309017 + 0.535233i) q^{3} +(-0.309017 - 0.535233i) q^{4} +(-0.330792 + 0.190983i) q^{6} +(-1.40126 + 0.809017i) q^{7} -1.00000i q^{8} +(0.309017 + 0.535233i) q^{9} +(-0.866025 - 0.500000i) q^{11} +0.381966 q^{12} -1.00000 q^{14} +0.381966i q^{18} +(-0.535233 + 0.309017i) q^{19} -1.00000i q^{21} +(-0.309017 - 0.535233i) q^{22} +(-0.809017 + 1.40126i) q^{23} +(0.535233 + 0.309017i) q^{24} -1.00000 q^{25} -1.00000 q^{27} +(0.866025 + 0.500000i) q^{28} +(-0.866025 + 0.500000i) q^{32} +(0.535233 - 0.309017i) q^{33} +(0.190983 - 0.330792i) q^{36} -0.381966 q^{38} +(-1.40126 - 0.809017i) q^{41} +(0.309017 - 0.535233i) q^{42} +0.618034i q^{44} +(-0.866025 + 0.500000i) q^{46} +(0.809017 - 1.40126i) q^{49} +(-0.535233 - 0.309017i) q^{50} +0.618034 q^{53} +(-0.535233 - 0.309017i) q^{54} +(0.809017 + 1.40126i) q^{56} -0.381966i q^{57} +(-0.866025 - 0.500000i) q^{63} -0.618034 q^{64} +0.381966 q^{66} +(-0.500000 - 0.866025i) q^{69} +(0.535233 - 0.309017i) q^{72} +0.618034i q^{73} +(0.309017 - 0.535233i) q^{75} +(0.330792 + 0.190983i) q^{76} +1.61803 q^{77} +(-0.500000 - 0.866025i) q^{82} +1.61803i q^{83} +(-0.535233 + 0.309017i) q^{84} +(-0.500000 + 0.866025i) q^{88} +1.00000 q^{92} -0.618034i q^{96} +(0.866025 - 0.500000i) q^{98} -0.618034i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 2 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 2 q^{4} - 2 q^{9} + 12 q^{12} - 8 q^{14} + 2 q^{22} - 2 q^{23} - 8 q^{25} - 8 q^{27} + 6 q^{36} - 12 q^{38} - 2 q^{42} + 2 q^{49} - 4 q^{53} + 2 q^{56} + 4 q^{64} + 12 q^{66} - 4 q^{69} - 2 q^{75} + 4 q^{77} - 4 q^{82} - 4 q^{88} + 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times\).

\(n\) \(508\) \(1354\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.535233 + 0.309017i 0.535233 + 0.309017i 0.743145 0.669131i \(-0.233333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(3\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(4\) −0.309017 0.535233i −0.309017 0.535233i
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(7\) −1.40126 + 0.809017i −1.40126 + 0.809017i −0.994522 0.104528i \(-0.966667\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(10\) 0 0
\(11\) −0.866025 0.500000i −0.866025 0.500000i
\(12\) 0.381966 0.381966
\(13\) 0 0
\(14\) −1.00000 −1.00000
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0.381966i 0.381966i
\(19\) −0.535233 + 0.309017i −0.535233 + 0.309017i −0.743145 0.669131i \(-0.766667\pi\)
0.207912 + 0.978148i \(0.433333\pi\)
\(20\) 0 0
\(21\) 1.00000i 1.00000i
\(22\) −0.309017 0.535233i −0.309017 0.535233i
\(23\) −0.809017 + 1.40126i −0.809017 + 1.40126i 0.104528 + 0.994522i \(0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(24\) 0.535233 + 0.309017i 0.535233 + 0.309017i
\(25\) −1.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(33\) 0.535233 0.309017i 0.535233 0.309017i
\(34\) 0 0
\(35\) 0 0
\(36\) 0.190983 0.330792i 0.190983 0.330792i
\(37\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(38\) −0.381966 −0.381966
\(39\) 0 0
\(40\) 0 0
\(41\) −1.40126 0.809017i −1.40126 0.809017i −0.406737 0.913545i \(-0.633333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(42\) 0.309017 0.535233i 0.309017 0.535233i
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) 0.618034i 0.618034i
\(45\) 0 0
\(46\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0.809017 1.40126i 0.809017 1.40126i
\(50\) −0.535233 0.309017i −0.535233 0.309017i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(54\) −0.535233 0.309017i −0.535233 0.309017i
\(55\) 0 0
\(56\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(57\) 0.381966i 0.381966i
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) −0.866025 0.500000i −0.866025 0.500000i
\(64\) −0.618034 −0.618034
\(65\) 0 0
\(66\) 0.381966 0.381966
\(67\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(68\) 0 0
\(69\) −0.500000 0.866025i −0.500000 0.866025i
\(70\) 0 0
\(71\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) 0.535233 0.309017i 0.535233 0.309017i
\(73\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(74\) 0 0
\(75\) 0.309017 0.535233i 0.309017 0.535233i
\(76\) 0.330792 + 0.190983i 0.330792 + 0.190983i
\(77\) 1.61803 1.61803
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.500000 0.866025i −0.500000 0.866025i
\(83\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(84\) −0.535233 + 0.309017i −0.535233 + 0.309017i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 1.00000
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.618034i 0.618034i
\(97\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(98\) 0.866025 0.500000i 0.866025 0.500000i
\(99\) 0.618034i 0.618034i
\(100\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.330792 + 0.190983i 0.330792 + 0.190983i
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(109\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(114\) 0.118034 0.204441i 0.118034 0.204441i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0.866025 0.500000i 0.866025 0.500000i
\(124\) 0 0
\(125\) 0 0
\(126\) −0.309017 0.535233i −0.309017 0.535233i
\(127\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) 0.535233 + 0.309017i 0.535233 + 0.309017i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) −0.330792 0.190983i −0.330792 0.190983i
\(133\) 0.500000 0.866025i 0.500000 0.866025i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(138\) 0.618034i 0.618034i
\(139\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(147\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(148\) 0 0
\(149\) 1.40126 0.809017i 1.40126 0.809017i 0.406737 0.913545i \(-0.366667\pi\)
0.994522 + 0.104528i \(0.0333333\pi\)
\(150\) 0.330792 0.190983i 0.330792 0.190983i
\(151\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(153\) 0 0
\(154\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(155\) 0 0
\(156\) 0 0
\(157\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(158\) 0 0
\(159\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(160\) 0 0
\(161\) 2.61803i 2.61803i
\(162\) 0 0
\(163\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(164\) 1.00000i 1.00000i
\(165\) 0 0
\(166\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(167\) −0.535233 0.309017i −0.535233 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(168\) −1.00000 −1.00000
\(169\) 0 0
\(170\) 0 0
\(171\) −0.330792 0.190983i −0.330792 0.190983i
\(172\) 0 0
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) 1.40126 0.809017i 1.40126 0.809017i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(180\) 0 0
\(181\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.40126 + 0.809017i 1.40126 + 0.809017i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.40126 0.809017i 1.40126 0.809017i
\(190\) 0 0
\(191\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(192\) 0.190983 0.330792i 0.190983 0.330792i
\(193\) 1.40126 + 0.809017i 1.40126 + 0.809017i 0.994522 0.104528i \(-0.0333333\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.00000 −1.00000
\(197\) 0.535233 + 0.309017i 0.535233 + 0.309017i 0.743145 0.669131i \(-0.233333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(198\) 0.190983 0.330792i 0.190983 0.330792i
\(199\) 0.309017 + 0.535233i 0.309017 + 0.535233i 0.978148 0.207912i \(-0.0666667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(200\) 1.00000i 1.00000i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(207\) −1.00000 −1.00000
\(208\) 0 0
\(209\) 0.618034 0.618034
\(210\) 0 0
\(211\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) −0.190983 0.330792i −0.190983 0.330792i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000i 1.00000i
\(217\) 0 0
\(218\) 0.190983 0.330792i 0.190983 0.330792i
\(219\) −0.330792 0.190983i −0.330792 0.190983i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(224\) 0.809017 1.40126i 0.809017 1.40126i
\(225\) −0.309017 0.535233i −0.309017 0.535233i
\(226\) 0.381966i 0.381966i
\(227\) −0.535233 + 0.309017i −0.535233 + 0.309017i −0.743145 0.669131i \(-0.766667\pi\)
0.207912 + 0.978148i \(0.433333\pi\)
\(228\) −0.204441 + 0.118034i −0.204441 + 0.118034i
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(240\) 0 0
\(241\) −1.40126 + 0.809017i −1.40126 + 0.809017i −0.994522 0.104528i \(-0.966667\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(242\) 0.618034i 0.618034i
\(243\) −0.500000 0.866025i −0.500000 0.866025i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.618034 0.618034
\(247\) 0 0
\(248\) 0 0
\(249\) −0.866025 0.500000i −0.866025 0.500000i
\(250\) 0 0
\(251\) −0.809017 1.40126i −0.809017 1.40126i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 0.994522i \(-0.466667\pi\)
\(252\) 0.618034i 0.618034i
\(253\) 1.40126 0.809017i 1.40126 0.809017i
\(254\) 0 0
\(255\) 0 0
\(256\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(257\) −0.809017 + 1.40126i −0.809017 + 1.40126i 0.104528 + 0.994522i \(0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) −0.309017 0.535233i −0.309017 0.535233i
\(265\) 0 0
\(266\) 0.535233 0.309017i 0.535233 0.309017i
\(267\) 0 0
\(268\) 0 0
\(269\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(270\) 0 0
\(271\) −0.535233 0.309017i −0.535233 0.309017i 0.207912 0.978148i \(-0.433333\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(276\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(282\) 0 0
\(283\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.61803 2.61803
\(288\) −0.535233 0.309017i −0.535233 0.309017i
\(289\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0.330792 0.190983i 0.330792 0.190983i
\(293\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(294\) 0.618034i 0.618034i
\(295\) 0 0
\(296\) 0 0
\(297\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(298\) 1.00000 1.00000
\(299\) 0 0
\(300\) −0.381966 −0.381966
\(301\) 0 0
\(302\) −0.618034 + 1.07047i −0.618034 + 1.07047i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −0.500000 0.866025i −0.500000 0.866025i
\(309\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(310\) 0 0
\(311\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(312\) 0 0
\(313\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(314\) −0.866025 0.500000i −0.866025 0.500000i
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) −0.204441 + 0.118034i −0.204441 + 0.118034i
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0.809017 1.40126i 0.809017 1.40126i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0.330792 + 0.190983i 0.330792 + 0.190983i
\(328\) −0.809017 + 1.40126i −0.809017 + 1.40126i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 0.866025 0.500000i 0.866025 0.500000i
\(333\) 0 0
\(334\) −0.190983 0.330792i −0.190983 0.330792i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0.381966 0.381966
\(340\) 0 0
\(341\) 0 0
\(342\) −0.118034 0.204441i −0.118034 0.204441i
\(343\) 1.00000i 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 1.40126 + 0.809017i 1.40126 + 0.809017i 0.994522 0.104528i \(-0.0333333\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(350\) 1.00000 1.00000
\(351\) 0 0
\(352\) 1.00000 1.00000
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.07047 0.618034i 1.07047 0.618034i
\(359\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(360\) 0 0
\(361\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(362\) −0.330792 0.190983i −0.330792 0.190983i
\(363\) −0.618034 −0.618034
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(368\) 0 0
\(369\) 1.00000i 1.00000i
\(370\) 0 0
\(371\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 1.00000 1.00000
\(379\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.381966i 0.381966i
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(385\) 0 0
\(386\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(387\) 0 0
\(388\) 0 0
\(389\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.40126 0.809017i −1.40126 0.809017i
\(393\) 0 0
\(394\) 0.190983 + 0.330792i 0.190983 + 0.330792i
\(395\) 0 0
\(396\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(397\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0.381966i 0.381966i
\(399\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(400\) 0 0
\(401\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.500000 0.866025i −0.500000 0.866025i
\(413\) 0 0
\(414\) −0.535233 0.309017i −0.535233 0.309017i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0.330792 + 0.190983i 0.330792 + 0.190983i
\(419\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0.618034i 0.618034i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.40126 0.809017i −1.40126 0.809017i −0.406737 0.913545i \(-0.633333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(432\) 0 0
\(433\) 0.309017 + 0.535233i 0.309017 + 0.535233i 0.978148 0.207912i \(-0.0666667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(437\) 1.00000i 1.00000i
\(438\) −0.118034 0.204441i −0.118034 0.204441i
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0 0
\(443\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.00000i 1.00000i
\(448\) 0.866025 0.500000i 0.866025 0.500000i
\(449\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(450\) 0.381966i 0.381966i
\(451\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(452\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(453\) −1.07047 0.618034i −1.07047 0.618034i
\(454\) −0.381966 −0.381966
\(455\) 0 0
\(456\) −0.381966 −0.381966
\(457\) 0.535233 + 0.309017i 0.535233 + 0.309017i 0.743145 0.669131i \(-0.233333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.535233 + 0.309017i −0.535233 + 0.309017i −0.743145 0.669131i \(-0.766667\pi\)
0.207912 + 0.978148i \(0.433333\pi\)
\(462\) −0.535233 + 0.309017i −0.535233 + 0.309017i
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.500000 0.866025i 0.500000 0.866025i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.535233 0.309017i 0.535233 0.309017i
\(476\) 0 0
\(477\) 0.190983 + 0.330792i 0.190983 + 0.330792i
\(478\) 0.190983 0.330792i 0.190983 0.330792i
\(479\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1.00000 −1.00000
\(483\) 1.40126 + 0.809017i 1.40126 + 0.809017i
\(484\) 0.309017 0.535233i 0.309017 0.535233i
\(485\) 0 0
\(486\) 0.618034i 0.618034i
\(487\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) −0.535233 0.309017i −0.535233 0.309017i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −0.309017 0.535233i −0.309017 0.535233i
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0.330792 0.190983i 0.330792 0.190983i
\(502\) 1.00000i 1.00000i
\(503\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(505\) 0 0
\(506\) 1.00000 1.00000
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) −0.500000 0.866025i −0.500000 0.866025i
\(512\) 0 0
\(513\) 0.535233 0.309017i 0.535233 0.309017i
\(514\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(522\) 0 0
\(523\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 1.00000i 1.00000i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.809017 1.40126i −0.809017 1.40126i
\(530\) 0 0
\(531\) 0 0
\(532\) −0.618034 −0.618034
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.618034 + 1.07047i 0.618034 + 1.07047i
\(538\) 1.00000i 1.00000i
\(539\) −1.40126 + 0.809017i −1.40126 + 0.809017i
\(540\) 0 0
\(541\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(542\) −0.190983 0.330792i −0.190983 0.330792i
\(543\) 0.190983 0.330792i 0.190983 0.330792i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(551\) 0 0
\(552\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.40126 + 0.809017i 1.40126 + 0.809017i 0.994522 0.104528i \(-0.0333333\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0.381966 0.381966
\(574\) 1.40126 + 0.809017i 1.40126 + 0.809017i
\(575\) 0.809017 1.40126i 0.809017 1.40126i
\(576\) −0.190983 0.330792i −0.190983 0.330792i
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −0.535233 + 0.309017i −0.535233 + 0.309017i
\(579\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(580\) 0 0
\(581\) −1.30902 2.26728i −1.30902 2.26728i
\(582\) 0 0
\(583\) −0.535233 0.309017i −0.535233 0.309017i
\(584\) 0.618034 0.618034
\(585\) 0 0
\(586\) 1.23607 1.23607
\(587\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(588\) 0.309017 0.535233i 0.309017 0.535233i
\(589\) 0 0
\(590\) 0 0
\(591\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(592\) 0 0
\(593\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(594\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(595\) 0 0
\(596\) −0.866025 0.500000i −0.866025 0.500000i
\(597\) −0.381966 −0.381966
\(598\) 0 0
\(599\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(600\) −0.535233 0.309017i −0.535233 0.309017i
\(601\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.07047 0.618034i 1.07047 0.618034i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 0.309017 0.535233i 0.309017 0.535233i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.535233 + 0.309017i 0.535233 + 0.309017i 0.743145 0.669131i \(-0.233333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(614\) −0.618034 + 1.07047i −0.618034 + 1.07047i
\(615\) 0 0
\(616\) 1.61803i 1.61803i
\(617\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) −0.535233 + 0.309017i −0.535233 + 0.309017i
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0.809017 1.40126i 0.809017 1.40126i
\(622\) −0.330792 0.190983i −0.330792 0.190983i
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 1.00000
\(626\) −0.866025 0.500000i −0.866025 0.500000i
\(627\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(628\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0.236068 0.236068
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.809017 1.40126i −0.809017 1.40126i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 0.994522i \(-0.466667\pi\)
\(642\) 0 0
\(643\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(644\) −1.40126 + 0.809017i −1.40126 + 0.809017i
\(645\) 0 0
\(646\) 0 0
\(647\) 0.309017 0.535233i 0.309017 0.535233i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0.118034 + 0.204441i 0.118034 + 0.204441i
\(655\) 0 0
\(656\) 0 0
\(657\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1.61803 1.61803
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0.381966i 0.381966i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(673\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 1.00000 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0.204441 + 0.118034i 0.204441 + 0.118034i
\(679\) 0 0
\(680\) 0 0
\(681\) 0.381966i 0.381966i
\(682\) 0 0
\(683\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(684\) 0.236068i 0.236068i
\(685\) 0 0
\(686\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) 0 0
\(693\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(699\) 0 0
\(700\) −0.866025 0.500000i −0.866025 0.500000i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.535233 + 0.309017i 0.535233 + 0.309017i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.23607 −1.23607
\(717\) 0.330792 + 0.190983i 0.330792 + 0.190983i
\(718\) 0.500000 0.866025i 0.500000 0.866025i
\(719\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) −2.26728 + 1.30902i −2.26728 + 1.30902i
\(722\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(723\) 1.00000i 1.00000i
\(724\) 0.190983 + 0.330792i 0.190983 + 0.330792i
\(725\) 0 0
\(726\) −0.330792 0.190983i −0.330792 0.190983i
\(727\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(728\) 0 0
\(729\) 0.618034 0.618034
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(734\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(735\) 0 0
\(736\) 1.61803i 1.61803i
\(737\) 0 0
\(738\) 0.309017 0.535233i 0.309017 0.535233i
\(739\) 1.40126 + 0.809017i 1.40126 + 0.809017i 0.994522 0.104528i \(-0.0333333\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.618034 −0.618034
\(743\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.309017 0.535233i 0.309017 0.535233i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(752\) 0 0
\(753\) 1.00000 1.00000
\(754\) 0 0
\(755\) 0 0
\(756\) −0.866025 0.500000i −0.866025 0.500000i
\(757\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(758\) 0 0
\(759\) 1.00000i 1.00000i
\(760\) 0 0
\(761\) −1.40126 + 0.809017i −1.40126 + 0.809017i −0.994522 0.104528i \(-0.966667\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(762\) 0 0
\(763\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(764\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.618034 −0.618034
\(769\) −1.40126 0.809017i −1.40126 0.809017i −0.406737 0.913545i \(-0.633333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(770\) 0 0
\(771\) −0.500000 0.866025i −0.500000 0.866025i
\(772\) 1.00000i 1.00000i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −0.330792 0.190983i −0.330792 0.190983i
\(779\) 1.00000 1.00000
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.40126 + 0.809017i −1.40126 + 0.809017i −0.994522 0.104528i \(-0.966667\pi\)
−0.406737 + 0.913545i \(0.633333\pi\)
\(788\) 0.381966i 0.381966i
\(789\) 0 0
\(790\) 0 0
\(791\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(792\) −0.618034 −0.618034
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0.190983 0.330792i 0.190983 0.330792i
\(797\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 0.381966i 0.381966i
\(799\) 0 0
\(800\) 0.866025 0.500000i 0.866025 0.500000i
\(801\) 0 0
\(802\) 0 0
\(803\) 0.309017 0.535233i 0.309017 0.535233i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.00000 −1.00000
\(808\) 0 0
\(809\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(812\) 0 0
\(813\) 0.330792 0.190983i 0.330792 0.190983i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.23607 −1.23607
\(819\) 0 0
\(820\) 0 0
\(821\) 1.73205 + 1.00000i 1.73205 + 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(822\) 0 0
\(823\) −0.809017 1.40126i −0.809017 1.40126i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 0.994522i \(-0.466667\pi\)
\(824\) 1.61803i 1.61803i
\(825\) −0.535233 + 0.309017i −0.535233 + 0.309017i
\(826\) 0 0
\(827\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(828\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(829\) −0.809017 + 1.40126i −0.809017 + 1.40126i 0.104528 + 0.994522i \(0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −0.190983 0.330792i −0.190983 0.330792i
\(837\) 0 0
\(838\) 0.866025 0.500000i 0.866025 0.500000i
\(839\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 0 0
\(841\) −0.500000 0.866025i −0.500000 0.866025i
\(842\) 0 0
\(843\) −0.330792 0.190983i −0.330792 0.190983i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.40126 0.809017i −1.40126 0.809017i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(860\) 0 0
\(861\) −0.809017 + 1.40126i −0.809017 + 1.40126i
\(862\) −0.500000 0.866025i −0.500000 0.866025i
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0.866025 0.500000i 0.866025 0.500000i
\(865\) 0 0
\(866\) 0.381966i 0.381966i
\(867\) −0.309017 0.535233i −0.309017 0.535233i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.618034 −0.618034
\(873\) 0 0
\(874\) 0.309017 0.535233i 0.309017 0.535233i
\(875\) 0 0
\(876\) 0.236068i 0.236068i
\(877\) −0.535233 + 0.309017i −0.535233 + 0.309017i −0.743145 0.669131i \(-0.766667\pi\)
0.207912 + 0.978148i \(0.433333\pi\)
\(878\) 0 0
\(879\) 1.23607i 1.23607i
\(880\) 0 0
\(881\) 0.309017 0.535233i 0.309017 0.535233i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(882\) 0.535233 + 0.309017i 0.535233 + 0.309017i
\(883\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.866025 0.500000i −0.866025 0.500000i
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(895\) 0 0
\(896\) −1.00000 −1.00000
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.190983 + 0.330792i −0.190983 + 0.330792i
\(901\) 0 0
\(902\) 1.00000i 1.00000i
\(903\) 0 0
\(904\) −0.535233 + 0.309017i −0.535233 + 0.309017i
\(905\) 0 0
\(906\) −0.381966 0.661585i −0.381966 0.661585i
\(907\) −0.809017 + 1.40126i −0.809017 + 1.40126i 0.104528 + 0.994522i \(0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(908\) 0.330792 + 0.190983i 0.330792 + 0.190983i
\(909\) 0 0
\(910\) 0 0
\(911\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(912\) 0 0
\(913\) 0.809017 1.40126i 0.809017 1.40126i
\(914\) 0.190983 + 0.330792i 0.190983 + 0.330792i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(920\) 0 0
\(921\) −1.07047 0.618034i −1.07047 0.618034i
\(922\) −0.381966 −0.381966
\(923\) 0 0
\(924\) 0.618034 0.618034
\(925\) 0 0
\(926\) 0 0
\(927\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(928\) 0 0
\(929\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(930\) 0 0
\(931\) 1.00000i 1.00000i
\(932\) 0 0
\(933\) 0.190983 0.330792i 0.190983 0.330792i
\(934\) −1.07047 0.618034i −1.07047 0.618034i
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0.500000 0.866025i 0.500000 0.866025i
\(940\) 0 0
\(941\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(942\) 0.535233 0.309017i 0.535233 0.309017i
\(943\) 2.26728 1.30902i 2.26728 1.30902i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.381966 0.381966
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(954\) 0.236068i 0.236068i
\(955\) 0 0
\(956\) −0.330792 + 0.190983i −0.330792 + 0.190983i
\(957\) 0 0
\(958\) −0.618034 1.07047i −0.618034 1.07047i
\(959\) 0 0
\(960\) 0 0
\(961\) −1.00000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(965\) 0 0
\(966\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(967\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(968\) 0.866025 0.500000i 0.866025 0.500000i
\(969\) 0 0
\(970\) 0 0
\(971\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(972\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.330792 0.190983i 0.330792 0.190983i
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −0.500000 0.866025i −0.500000 0.866025i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0.618034i 0.618034i
\(997\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1859.1.k.c.1374.3 8
11.10 odd 2 inner 1859.1.k.c.1374.2 8
13.2 odd 12 1859.1.i.a.868.2 4
13.3 even 3 inner 1859.1.k.c.1836.2 8
13.4 even 6 1859.1.c.c.846.3 4
13.5 odd 4 1859.1.i.a.1330.2 4
13.6 odd 12 143.1.d.b.142.1 yes 2
13.7 odd 12 143.1.d.a.142.2 2
13.8 odd 4 1859.1.i.b.1330.1 4
13.9 even 3 1859.1.c.c.846.2 4
13.10 even 6 inner 1859.1.k.c.1836.3 8
13.11 odd 12 1859.1.i.b.868.1 4
13.12 even 2 inner 1859.1.k.c.1374.2 8
39.20 even 12 1287.1.g.b.1000.1 2
39.32 even 12 1287.1.g.a.1000.2 2
52.7 even 12 2288.1.m.b.2001.1 2
52.19 even 12 2288.1.m.a.2001.1 2
65.7 even 12 3575.1.c.d.3574.3 4
65.19 odd 12 3575.1.h.e.2001.2 2
65.32 even 12 3575.1.c.c.3574.2 4
65.33 even 12 3575.1.c.d.3574.2 4
65.58 even 12 3575.1.c.c.3574.3 4
65.59 odd 12 3575.1.h.f.2001.1 2
143.6 even 60 1573.1.l.b.844.1 4
143.7 even 60 1573.1.l.a.1546.1 4
143.10 odd 6 inner 1859.1.k.c.1836.2 8
143.19 even 60 1573.1.l.d.233.1 4
143.20 odd 60 1573.1.l.b.766.1 4
143.21 even 4 1859.1.i.a.1330.2 4
143.32 even 12 143.1.d.a.142.2 2
143.43 odd 6 1859.1.c.c.846.2 4
143.46 even 60 1573.1.l.c.766.1 4
143.54 even 12 1859.1.i.b.868.1 4
143.58 odd 60 1573.1.l.a.233.1 4
143.59 odd 60 1573.1.l.d.1546.1 4
143.71 odd 60 1573.1.l.c.844.1 4
143.72 even 60 1573.1.l.c.844.1 4
143.76 even 12 1859.1.i.a.868.2 4
143.84 even 60 1573.1.l.d.1546.1 4
143.85 even 60 1573.1.l.a.233.1 4
143.87 odd 6 1859.1.c.c.846.3 4
143.97 odd 60 1573.1.l.c.766.1 4
143.98 even 12 143.1.d.b.142.1 yes 2
143.109 even 4 1859.1.i.b.1330.1 4
143.120 odd 6 inner 1859.1.k.c.1836.3 8
143.123 even 60 1573.1.l.b.766.1 4
143.124 odd 60 1573.1.l.d.233.1 4
143.136 odd 60 1573.1.l.a.1546.1 4
143.137 odd 60 1573.1.l.b.844.1 4
143.142 odd 2 CM 1859.1.k.c.1374.3 8
429.32 odd 12 1287.1.g.b.1000.1 2
429.98 odd 12 1287.1.g.a.1000.2 2
572.175 odd 12 2288.1.m.b.2001.1 2
572.527 odd 12 2288.1.m.a.2001.1 2
715.32 odd 12 3575.1.c.d.3574.3 4
715.98 odd 12 3575.1.c.c.3574.3 4
715.318 odd 12 3575.1.c.d.3574.2 4
715.384 even 12 3575.1.h.e.2001.2 2
715.527 odd 12 3575.1.c.c.3574.2 4
715.604 even 12 3575.1.h.f.2001.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
143.1.d.a.142.2 2 13.7 odd 12
143.1.d.a.142.2 2 143.32 even 12
143.1.d.b.142.1 yes 2 13.6 odd 12
143.1.d.b.142.1 yes 2 143.98 even 12
1287.1.g.a.1000.2 2 39.32 even 12
1287.1.g.a.1000.2 2 429.98 odd 12
1287.1.g.b.1000.1 2 39.20 even 12
1287.1.g.b.1000.1 2 429.32 odd 12
1573.1.l.a.233.1 4 143.58 odd 60
1573.1.l.a.233.1 4 143.85 even 60
1573.1.l.a.1546.1 4 143.7 even 60
1573.1.l.a.1546.1 4 143.136 odd 60
1573.1.l.b.766.1 4 143.20 odd 60
1573.1.l.b.766.1 4 143.123 even 60
1573.1.l.b.844.1 4 143.6 even 60
1573.1.l.b.844.1 4 143.137 odd 60
1573.1.l.c.766.1 4 143.46 even 60
1573.1.l.c.766.1 4 143.97 odd 60
1573.1.l.c.844.1 4 143.71 odd 60
1573.1.l.c.844.1 4 143.72 even 60
1573.1.l.d.233.1 4 143.19 even 60
1573.1.l.d.233.1 4 143.124 odd 60
1573.1.l.d.1546.1 4 143.59 odd 60
1573.1.l.d.1546.1 4 143.84 even 60
1859.1.c.c.846.2 4 13.9 even 3
1859.1.c.c.846.2 4 143.43 odd 6
1859.1.c.c.846.3 4 13.4 even 6
1859.1.c.c.846.3 4 143.87 odd 6
1859.1.i.a.868.2 4 13.2 odd 12
1859.1.i.a.868.2 4 143.76 even 12
1859.1.i.a.1330.2 4 13.5 odd 4
1859.1.i.a.1330.2 4 143.21 even 4
1859.1.i.b.868.1 4 13.11 odd 12
1859.1.i.b.868.1 4 143.54 even 12
1859.1.i.b.1330.1 4 13.8 odd 4
1859.1.i.b.1330.1 4 143.109 even 4
1859.1.k.c.1374.2 8 11.10 odd 2 inner
1859.1.k.c.1374.2 8 13.12 even 2 inner
1859.1.k.c.1374.3 8 1.1 even 1 trivial
1859.1.k.c.1374.3 8 143.142 odd 2 CM
1859.1.k.c.1836.2 8 13.3 even 3 inner
1859.1.k.c.1836.2 8 143.10 odd 6 inner
1859.1.k.c.1836.3 8 13.10 even 6 inner
1859.1.k.c.1836.3 8 143.120 odd 6 inner
2288.1.m.a.2001.1 2 52.19 even 12
2288.1.m.a.2001.1 2 572.527 odd 12
2288.1.m.b.2001.1 2 52.7 even 12
2288.1.m.b.2001.1 2 572.175 odd 12
3575.1.c.c.3574.2 4 65.32 even 12
3575.1.c.c.3574.2 4 715.527 odd 12
3575.1.c.c.3574.3 4 65.58 even 12
3575.1.c.c.3574.3 4 715.98 odd 12
3575.1.c.d.3574.2 4 65.33 even 12
3575.1.c.d.3574.2 4 715.318 odd 12
3575.1.c.d.3574.3 4 65.7 even 12
3575.1.c.d.3574.3 4 715.32 odd 12
3575.1.h.e.2001.2 2 65.19 odd 12
3575.1.h.e.2001.2 2 715.384 even 12
3575.1.h.f.2001.1 2 65.59 odd 12
3575.1.h.f.2001.1 2 715.604 even 12