Properties

Label 3528.1.co.a.587.7
Level $3528$
Weight $1$
Character 3528.587
Analytic conductor $1.761$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,1,Mod(587,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.587");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.co (of order \(6\), degree \(2\), 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: \(1.76070136457\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} + \cdots)\)

Embedding invariants

Embedding label 587.7
Root \(0.130526 + 0.991445i\) of defining polynomial
Character \(\chi\) \(=\) 3528.587
Dual form 3528.1.co.a.2939.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.608761 + 0.793353i) q^{3} +(0.500000 + 0.866025i) q^{4} +(0.130526 + 0.991445i) q^{6} +1.00000i q^{8} +(-0.258819 + 0.965926i) q^{9} +O(q^{10})\) \(q+(0.866025 + 0.500000i) q^{2} +(0.608761 + 0.793353i) q^{3} +(0.500000 + 0.866025i) q^{4} +(0.130526 + 0.991445i) q^{6} +1.00000i q^{8} +(-0.258819 + 0.965926i) q^{9} +(0.448288 + 0.258819i) q^{11} +(-0.382683 + 0.923880i) q^{12} +(-0.500000 + 0.866025i) q^{16} +0.261052 q^{17} +(-0.707107 + 0.707107i) q^{18} -1.21752i q^{19} +(0.258819 + 0.448288i) q^{22} +(-0.793353 + 0.608761i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-0.923880 + 0.382683i) q^{27} +(-0.866025 + 0.500000i) q^{32} +(0.0675653 + 0.513210i) q^{33} +(0.226078 + 0.130526i) q^{34} +(-0.965926 + 0.258819i) q^{36} +(0.608761 - 1.05441i) q^{38} +(-0.991445 - 1.71723i) q^{41} +(0.965926 - 1.67303i) q^{43} +0.517638i q^{44} +(-0.991445 + 0.130526i) q^{48} +(-0.866025 + 0.500000i) q^{50} +(0.158919 + 0.207107i) q^{51} +(-0.991445 - 0.130526i) q^{54} +(0.965926 - 0.741181i) q^{57} +(0.793353 + 1.37413i) q^{59} -1.00000 q^{64} +(-0.198092 + 0.478235i) q^{66} +(0.866025 + 1.50000i) q^{67} +(0.130526 + 0.226078i) q^{68} +(-0.965926 - 0.258819i) q^{72} -1.58671i q^{73} +(-0.991445 + 0.130526i) q^{75} +(1.05441 - 0.608761i) q^{76} +(-0.866025 - 0.500000i) q^{81} -1.98289i q^{82} +(-0.923880 + 1.60021i) q^{83} +(1.67303 - 0.965926i) q^{86} +(-0.258819 + 0.448288i) q^{88} +0.765367 q^{89} +(-0.923880 - 0.382683i) q^{96} +(-1.71723 - 0.991445i) q^{97} +(-0.366025 + 0.366025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 8 q^{16} - 8 q^{25} - 16 q^{64} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(3\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(4\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(5\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(6\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(7\) 0 0
\(8\) 1.00000i 1.00000i
\(9\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(10\) 0 0
\(11\) 0.448288 + 0.258819i 0.448288 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(12\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(13\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(17\) 0.261052 0.261052 0.130526 0.991445i \(-0.458333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(18\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(19\) 1.21752i 1.21752i −0.793353 0.608761i \(-0.791667\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.258819 + 0.448288i 0.258819 + 0.448288i
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(28\) 0 0
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(33\) 0.0675653 + 0.513210i 0.0675653 + 0.513210i
\(34\) 0.226078 + 0.130526i 0.226078 + 0.130526i
\(35\) 0 0
\(36\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0.608761 1.05441i 0.608761 1.05441i
\(39\) 0 0
\(40\) 0 0
\(41\) −0.991445 1.71723i −0.991445 1.71723i −0.608761 0.793353i \(-0.708333\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(42\) 0 0
\(43\) 0.965926 1.67303i 0.965926 1.67303i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(44\) 0.517638i 0.517638i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(49\) 0 0
\(50\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(51\) 0.158919 + 0.207107i 0.158919 + 0.207107i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −0.991445 0.130526i −0.991445 0.130526i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.965926 0.741181i 0.965926 0.741181i
\(58\) 0 0
\(59\) 0.793353 + 1.37413i 0.793353 + 1.37413i 0.923880 + 0.382683i \(0.125000\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) −0.198092 + 0.478235i −0.198092 + 0.478235i
\(67\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(68\) 0.130526 + 0.226078i 0.130526 + 0.226078i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −0.965926 0.258819i −0.965926 0.258819i
\(73\) 1.58671i 1.58671i −0.608761 0.793353i \(-0.708333\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(74\) 0 0
\(75\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(76\) 1.05441 0.608761i 1.05441 0.608761i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.866025 0.500000i −0.866025 0.500000i
\(82\) 1.98289i 1.98289i
\(83\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.67303 0.965926i 1.67303 0.965926i
\(87\) 0 0
\(88\) −0.258819 + 0.448288i −0.258819 + 0.448288i
\(89\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −0.923880 0.382683i −0.923880 0.382683i
\(97\) −1.71723 0.991445i −1.71723 0.991445i −0.923880 0.382683i \(-0.875000\pi\)
−0.793353 0.608761i \(-0.791667\pi\)
\(98\) 0 0
\(99\) −0.366025 + 0.366025i −0.366025 + 0.366025i
\(100\) −1.00000 −1.00000
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0.0340742 + 0.258819i 0.0340742 + 0.258819i
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(108\) −0.793353 0.608761i −0.793353 0.608761i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(114\) 1.20711 0.158919i 1.20711 0.158919i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1.58671i 1.58671i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.366025 0.633975i −0.366025 0.633975i
\(122\) 0 0
\(123\) 0.758819 1.83195i 0.758819 1.83195i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −0.866025 0.500000i −0.866025 0.500000i
\(129\) 1.91532 0.252157i 1.91532 0.252157i
\(130\) 0 0
\(131\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(132\) −0.410670 + 0.315118i −0.410670 + 0.315118i
\(133\) 0 0
\(134\) 1.73205i 1.73205i
\(135\) 0 0
\(136\) 0.261052i 0.261052i
\(137\) 1.67303 + 0.965926i 1.67303 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) 0 0
\(139\) 1.37413 0.793353i 1.37413 0.793353i 0.382683 0.923880i \(-0.375000\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.707107 0.707107i −0.707107 0.707107i
\(145\) 0 0
\(146\) 0.793353 1.37413i 0.793353 1.37413i
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) −0.923880 0.382683i −0.923880 0.382683i
\(151\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 1.21752 1.21752
\(153\) −0.0675653 + 0.252157i −0.0675653 + 0.252157i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.500000 0.866025i −0.500000 0.866025i
\(163\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0.991445 1.71723i 0.991445 1.71723i
\(165\) 0 0
\(166\) −1.60021 + 0.923880i −1.60021 + 0.923880i
\(167\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(168\) 0 0
\(169\) 0.500000 0.866025i 0.500000 0.866025i
\(170\) 0 0
\(171\) 1.17604 + 0.315118i 1.17604 + 0.315118i
\(172\) 1.93185 1.93185
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.448288 + 0.258819i −0.448288 + 0.258819i
\(177\) −0.607206 + 1.46593i −0.607206 + 1.46593i
\(178\) 0.662827 + 0.382683i 0.662827 + 0.382683i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.117027 + 0.0675653i 0.117027 + 0.0675653i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) −0.608761 0.793353i −0.608761 0.793353i
\(193\) −0.965926 1.67303i −0.965926 1.67303i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(194\) −0.991445 1.71723i −0.991445 1.71723i
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −0.500000 + 0.133975i −0.500000 + 0.133975i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −0.866025 0.500000i −0.866025 0.500000i
\(201\) −0.662827 + 1.60021i −0.662827 + 1.60021i
\(202\) 0 0
\(203\) 0 0
\(204\) −0.0999004 + 0.241181i −0.0999004 + 0.241181i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.315118 0.545801i 0.315118 0.545801i
\(210\) 0 0
\(211\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.866025 1.50000i 0.866025 1.50000i
\(215\) 0 0
\(216\) −0.382683 0.923880i −0.382683 0.923880i
\(217\) 0 0
\(218\) 0 0
\(219\) 1.25882 0.965926i 1.25882 0.965926i
\(220\) 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 0
\(225\) −0.707107 0.707107i −0.707107 0.707107i
\(226\) −1.41421 −1.41421
\(227\) −0.991445 + 1.71723i −0.991445 + 1.71723i −0.382683 + 0.923880i \(0.625000\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(228\) 1.12484 + 0.465926i 1.12484 + 0.465926i
\(229\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.793353 + 1.37413i −0.793353 + 1.37413i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) −0.226078 0.130526i −0.226078 0.130526i 0.382683 0.923880i \(-0.375000\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(242\) 0.732051i 0.732051i
\(243\) −0.130526 0.991445i −0.130526 0.991445i
\(244\) 0 0
\(245\) 0 0
\(246\) 1.57313 1.20711i 1.57313 1.20711i
\(247\) 0 0
\(248\) 0 0
\(249\) −1.83195 + 0.241181i −1.83195 + 0.241181i
\(250\) 0 0
\(251\) 1.21752 1.21752 0.608761 0.793353i \(-0.291667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) 0.793353 + 1.37413i 0.793353 + 1.37413i 0.923880 + 0.382683i \(0.125000\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(258\) 1.78480 + 0.739288i 1.78480 + 0.739288i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0.765367i 0.765367i
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) −0.513210 + 0.0675653i −0.513210 + 0.0675653i
\(265\) 0 0
\(266\) 0 0
\(267\) 0.465926 + 0.607206i 0.465926 + 0.607206i
\(268\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −0.130526 + 0.226078i −0.130526 + 0.226078i
\(273\) 0 0
\(274\) 0.965926 + 1.67303i 0.965926 + 1.67303i
\(275\) −0.448288 + 0.258819i −0.448288 + 0.258819i
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 1.58671 1.58671
\(279\) 0 0
\(280\) 0 0
\(281\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(282\) 0 0
\(283\) 0.662827 0.382683i 0.662827 0.382683i −0.130526 0.991445i \(-0.541667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.258819 0.965926i −0.258819 0.965926i
\(289\) −0.931852 −0.931852
\(290\) 0 0
\(291\) −0.258819 1.96593i −0.258819 1.96593i
\(292\) 1.37413 0.793353i 1.37413 0.793353i
\(293\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.513210 0.0675653i −0.513210 0.0675653i
\(298\) 0 0
\(299\) 0 0
\(300\) −0.608761 0.793353i −0.608761 0.793353i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.05441 + 0.608761i 1.05441 + 0.608761i
\(305\) 0 0
\(306\) −0.184592 + 0.184592i −0.184592 + 0.184592i
\(307\) 0.261052i 0.261052i −0.991445 0.130526i \(-0.958333\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 0.226078 + 0.130526i 0.226078 + 0.130526i 0.608761 0.793353i \(-0.291667\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 1.37413 1.05441i 1.37413 1.05441i
\(322\) 0 0
\(323\) 0.317837i 0.317837i
\(324\) 1.00000i 1.00000i
\(325\) 0 0
\(326\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(327\) 0 0
\(328\) 1.71723 0.991445i 1.71723 0.991445i
\(329\) 0 0
\(330\) 0 0
\(331\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(332\) −1.84776 −1.84776
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(338\) 0.866025 0.500000i 0.866025 0.500000i
\(339\) −1.30656 0.541196i −1.30656 0.541196i
\(340\) 0 0
\(341\) 0 0
\(342\) 0.860919 + 0.860919i 0.860919 + 0.860919i
\(343\) 0 0
\(344\) 1.67303 + 0.965926i 1.67303 + 0.965926i
\(345\) 0 0
\(346\) 0 0
\(347\) −1.67303 + 0.965926i −1.67303 + 0.965926i −0.707107 + 0.707107i \(0.750000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(348\) 0 0
\(349\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.517638 −0.517638
\(353\) −0.608761 + 1.05441i −0.608761 + 1.05441i 0.382683 + 0.923880i \(0.375000\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(354\) −1.25882 + 0.965926i −1.25882 + 0.965926i
\(355\) 0 0
\(356\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −0.482362 −0.482362
\(362\) 0 0
\(363\) 0.280144 0.676327i 0.280144 0.676327i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) 0 0
\(369\) 1.91532 0.513210i 1.91532 0.513210i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0.0675653 + 0.117027i 0.0675653 + 0.117027i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.517638 −0.517638 −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(384\) −0.130526 0.991445i −0.130526 0.991445i
\(385\) 0 0
\(386\) 1.93185i 1.93185i
\(387\) 1.36603 + 1.36603i 1.36603 + 1.36603i
\(388\) 1.98289i 1.98289i
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(394\) 0 0
\(395\) 0 0
\(396\) −0.500000 0.133975i −0.500000 0.133975i
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.500000 0.866025i −0.500000 0.866025i
\(401\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(402\) −1.37413 + 1.05441i −1.37413 + 1.05441i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −0.207107 + 0.158919i −0.207107 + 0.158919i
\(409\) −1.71723 + 0.991445i −1.71723 + 0.991445i −0.793353 + 0.608761i \(0.791667\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(410\) 0 0
\(411\) 0.252157 + 1.91532i 0.252157 + 1.91532i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.46593 + 0.607206i 1.46593 + 0.607206i
\(418\) 0.545801 0.315118i 0.545801 0.315118i
\(419\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(420\) 0 0
\(421\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.130526 + 0.226078i −0.130526 + 0.226078i
\(426\) 0 0
\(427\) 0 0
\(428\) 1.50000 0.866025i 1.50000 0.866025i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0.130526 0.991445i 0.130526 0.991445i
\(433\) 1.21752i 1.21752i 0.793353 + 0.608761i \(0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1.57313 0.207107i 1.57313 0.207107i
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(450\) −0.258819 0.965926i −0.258819 0.965926i
\(451\) 1.02642i 1.02642i
\(452\) −1.22474 0.707107i −1.22474 0.707107i
\(453\) 0 0
\(454\) −1.71723 + 0.991445i −1.71723 + 0.991445i
\(455\) 0 0
\(456\) 0.741181 + 0.965926i 0.741181 + 0.965926i
\(457\) 0.258819 0.448288i 0.258819 0.448288i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(458\) 0 0
\(459\) −0.241181 + 0.0999004i −0.241181 + 0.0999004i
\(460\) 0 0
\(461\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0.965926 1.67303i 0.965926 1.67303i
\(467\) 0.261052 0.261052 0.130526 0.991445i \(-0.458333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.37413 + 0.793353i −1.37413 + 0.793353i
\(473\) 0.866025 0.500000i 0.866025 0.500000i
\(474\) 0 0
\(475\) 1.05441 + 0.608761i 1.05441 + 0.608761i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.130526 0.226078i −0.130526 0.226078i
\(483\) 0 0
\(484\) 0.366025 0.633975i 0.366025 0.633975i
\(485\) 0 0
\(486\) 0.382683 0.923880i 0.382683 0.923880i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0.860919 + 1.12197i 0.860919 + 1.12197i
\(490\) 0 0
\(491\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 1.96593 0.258819i 1.96593 0.258819i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −1.70711 0.707107i −1.70711 0.707107i
\(499\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.05441 + 0.608761i 1.05441 + 0.608761i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.991445 0.130526i 0.991445 0.130526i
\(508\) 0 0
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0.465926 + 1.12484i 0.465926 + 1.12484i
\(514\) 1.58671i 1.58671i
\(515\) 0 0
\(516\) 1.17604 + 1.53264i 1.17604 + 1.53264i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.58671 −1.58671 −0.793353 0.608761i \(-0.791667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(522\) 0 0
\(523\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(524\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −0.478235 0.198092i −0.478235 0.198092i
\(529\) 0.500000 0.866025i 0.500000 0.866025i
\(530\) 0 0
\(531\) −1.53264 + 0.410670i −1.53264 + 0.410670i
\(532\) 0 0
\(533\) 0 0
\(534\) 0.0999004 + 0.758819i 0.0999004 + 0.758819i
\(535\) 0 0
\(536\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.226078 + 0.130526i −0.226078 + 0.130526i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.258819 0.448288i 0.258819 0.448288i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(548\) 1.93185i 1.93185i
\(549\) 0 0
\(550\) −0.517638 −0.517638
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 1.37413 + 0.793353i 1.37413 + 0.793353i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0.0176381 + 0.133975i 0.0176381 + 0.133975i
\(562\) −0.707107 1.22474i −0.707107 1.22474i
\(563\) −0.608761 1.05441i −0.608761 1.05441i −0.991445 0.130526i \(-0.958333\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0.765367 0.765367
\(567\) 0 0
\(568\) 0 0
\(569\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0.258819 + 0.448288i 0.258819 + 0.448288i 0.965926 0.258819i \(-0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.258819 0.965926i 0.258819 0.965926i
\(577\) 1.21752i 1.21752i −0.793353 0.608761i \(-0.791667\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(578\) −0.807007 0.465926i −0.807007 0.465926i
\(579\) 0.739288 1.78480i 0.739288 1.78480i
\(580\) 0 0
\(581\) 0 0
\(582\) 0.758819 1.83195i 0.758819 1.83195i
\(583\) 0 0
\(584\) 1.58671 1.58671
\(585\) 0 0
\(586\) 0 0
\(587\) −0.991445 + 1.71723i −0.991445 + 1.71723i −0.382683 + 0.923880i \(0.625000\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(594\) −0.410670 0.315118i −0.410670 0.315118i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) −0.130526 0.991445i −0.130526 0.991445i
\(601\) −1.05441 0.608761i −1.05441 0.608761i −0.130526 0.991445i \(-0.541667\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(602\) 0 0
\(603\) −1.67303 + 0.448288i −1.67303 + 0.448288i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0.608761 + 1.05441i 0.608761 + 1.05441i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −0.252157 + 0.0675653i −0.252157 + 0.0675653i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0.130526 0.226078i 0.130526 0.226078i
\(615\) 0 0
\(616\) 0 0
\(617\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1.37413 0.793353i −1.37413 0.793353i −0.382683 0.923880i \(-0.625000\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.500000 0.866025i
\(626\) 0.130526 + 0.226078i 0.130526 + 0.226078i
\(627\) 0.624844 0.0822623i 0.624844 0.0822623i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(642\) 1.71723 0.226078i 1.71723 0.226078i
\(643\) −0.226078 + 0.130526i −0.226078 + 0.130526i −0.608761 0.793353i \(-0.708333\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.158919 0.275255i 0.158919 0.275255i
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0.500000 0.866025i 0.500000 0.866025i
\(649\) 0.821340i 0.821340i
\(650\) 0 0
\(651\) 0 0
\(652\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(653\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.98289 1.98289
\(657\) 1.53264 + 0.410670i 1.53264 + 0.410670i
\(658\) 0 0
\(659\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(660\) 0 0
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 1.22474 0.707107i 1.22474 0.707107i
\(663\) 0 0
\(664\) −1.60021 0.923880i −1.60021 0.923880i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(674\) 1.73205i 1.73205i
\(675\) 0.130526 0.991445i 0.130526 0.991445i
\(676\) 1.00000 1.00000
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) −0.860919 1.12197i −0.860919 1.12197i
\(679\) 0 0
\(680\) 0 0
\(681\) −1.96593 + 0.258819i −1.96593 + 0.258819i
\(682\) 0 0
\(683\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(684\) 0.315118 + 1.17604i 0.315118 + 1.17604i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0.965926 + 1.67303i 0.965926 + 1.67303i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.93185 −1.93185
\(695\) 0 0
\(696\) 0 0
\(697\) −0.258819 0.448288i −0.258819 0.448288i
\(698\) 0 0
\(699\) 1.53264 1.17604i 1.53264 1.17604i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.448288 0.258819i −0.448288 0.258819i
\(705\) 0 0
\(706\) −1.05441 + 0.608761i −1.05441 + 0.608761i
\(707\) 0 0
\(708\) −1.57313 + 0.207107i −1.57313 + 0.207107i
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.765367i 0.765367i
\(713\) 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 0
\(722\) −0.417738 0.241181i −0.417738 0.241181i
\(723\) −0.0340742 0.258819i −0.0340742 0.258819i
\(724\) 0 0
\(725\) 0 0
\(726\) 0.580775 0.445644i 0.580775 0.445644i
\(727\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(728\) 0 0
\(729\) 0.707107 0.707107i 0.707107 0.707107i
\(730\) 0 0
\(731\) 0.252157 0.436749i 0.252157 0.436749i
\(732\) 0 0
\(733\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.896575i 0.896575i
\(738\) 1.91532 + 0.513210i 1.91532 + 0.513210i
\(739\) −1.93185 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.30656 1.30656i −1.30656 1.30656i
\(748\) 0.135131i 0.135131i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0 0
\(753\) 0.741181 + 0.965926i 0.741181 + 0.965926i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −0.448288 0.258819i −0.448288 0.258819i
\(759\) 0 0
\(760\) 0 0
\(761\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.382683 0.923880i 0.382683 0.923880i
\(769\) −1.60021 + 0.923880i −1.60021 + 0.923880i −0.608761 + 0.793353i \(0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(770\) 0 0
\(771\) −0.607206 + 1.46593i −0.607206 + 1.46593i
\(772\) 0.965926 1.67303i 0.965926 1.67303i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0.500000 + 1.86603i 0.500000 + 1.86603i
\(775\) 0 0
\(776\) 0.991445 1.71723i 0.991445 1.71723i
\(777\) 0 0
\(778\) 0 0
\(779\) −2.09077 + 1.20711i −2.09077 + 1.20711i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) −0.607206 + 0.465926i −0.607206 + 0.465926i
\(787\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.366025 0.366025i −0.366025 0.366025i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 1.00000i
\(801\) −0.198092 + 0.739288i −0.198092 + 0.739288i
\(802\) −1.73205 −1.73205
\(803\) 0.410670 0.711301i 0.410670 0.711301i
\(804\) −1.71723 + 0.226078i −1.71723 + 0.226078i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.517638i 0.517638i −0.965926 0.258819i \(-0.916667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(810\) 0 0
\(811\) 1.98289i 1.98289i −0.130526 0.991445i \(-0.541667\pi\)
0.130526 0.991445i \(-0.458333\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −0.258819 + 0.0340742i −0.258819 + 0.0340742i
\(817\) −2.03696 1.17604i −2.03696 1.17604i
\(818\) −1.98289 −1.98289
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(822\) −0.739288 + 1.78480i −0.739288 + 1.78480i
\(823\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0 0
\(825\) −0.478235 0.198092i −0.478235 0.198092i
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0.965926 + 1.25882i 0.965926 + 1.25882i
\(835\) 0 0
\(836\) 0.630236 0.630236
\(837\) 0 0
\(838\) 1.84776i 1.84776i
\(839\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(840\) 0 0
\(841\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(842\) 0 0
\(843\) −0.184592 1.40211i −0.184592 1.40211i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(850\) −0.226078 + 0.130526i −0.226078 + 0.130526i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.73205 1.73205
\(857\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(858\) 0 0
\(859\) −1.71723 + 0.991445i −1.71723 + 0.991445i −0.793353 + 0.608761i \(0.791667\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0.608761 0.793353i 0.608761 0.793353i
\(865\) 0 0
\(866\) −0.608761 + 1.05441i −0.608761 + 1.05441i
\(867\) −0.567275 0.739288i −0.567275 0.739288i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.40211 1.40211i 1.40211 1.40211i
\(874\) 0 0
\(875\) 0 0
\(876\) 1.46593 + 0.607206i 1.46593 + 0.607206i
\(877\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(882\) 0 0
\(883\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.500000 0.866025i −0.500000 0.866025i
\(887\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.258819 0.448288i −0.258819 0.448288i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0.500000 0.866025i 0.500000 0.866025i
\(899\) 0 0
\(900\) 0.258819 0.965926i 0.258819 0.965926i
\(901\) 0 0
\(902\) 0.513210 0.888905i 0.513210 0.888905i
\(903\) 0 0
\(904\) −0.707107 1.22474i −0.707107 1.22474i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(908\) −1.98289 −1.98289
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(912\) 0.158919 + 1.20711i 0.158919 + 1.20711i
\(913\) −0.828328 + 0.478235i −0.828328 + 0.478235i
\(914\) 0.448288 0.258819i 0.448288 0.258819i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −0.258819 0.0340742i −0.258819 0.0340742i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0.207107 0.158919i 0.207107 0.158919i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.67303 0.965926i 1.67303 0.965926i
\(933\) 0 0
\(934\) 0.226078 + 0.130526i 0.226078 + 0.130526i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(938\) 0 0
\(939\) 0.0340742 + 0.258819i 0.0340742 + 0.258819i
\(940\) 0 0
\(941\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.58671 −1.58671
\(945\) 0 0
\(946\) 1.00000 1.00000
\(947\) 1.67303 + 0.965926i 1.67303 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.608761 + 1.05441i 0.608761 + 1.05441i
\(951\) 0 0
\(952\) 0 0
\(953\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.500000 0.866025i 0.500000 0.866025i
\(962\) 0 0
\(963\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(964\) 0.261052i 0.261052i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) 0.633975 0.366025i 0.633975 0.366025i
\(969\) 0.252157 0.193487i 0.252157 0.193487i
\(970\) 0 0
\(971\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(972\) 0.793353 0.608761i 0.793353 0.608761i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.448288 + 0.258819i −0.448288 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(978\) 0.184592 + 1.40211i 0.184592 + 1.40211i
\(979\) 0.343105 + 0.198092i 0.343105 + 0.198092i
\(980\) 0 0
\(981\) 0 0
\(982\) 1.00000 1.00000
\(983\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 1.83195 + 0.758819i 1.83195 + 0.758819i
\(985\) 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
\(993\) 1.40211 0.184592i 1.40211 0.184592i
\(994\) 0 0
\(995\) 0 0
\(996\) −1.12484 1.46593i −1.12484 1.46593i
\(997\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(998\) 1.73205i 1.73205i
\(999\) 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 3528.1.co.a.587.7 yes 16
7.2 even 3 3528.1.cm.a.227.1 16
7.3 odd 6 3528.1.u.a.803.2 16
7.4 even 3 3528.1.u.a.803.3 16
7.5 odd 6 3528.1.cm.a.227.4 16
7.6 odd 2 inner 3528.1.co.a.587.6 16
8.3 odd 2 CM 3528.1.co.a.587.7 yes 16
9.5 odd 6 inner 3528.1.co.a.2939.6 yes 16
56.3 even 6 3528.1.u.a.803.2 16
56.11 odd 6 3528.1.u.a.803.3 16
56.19 even 6 3528.1.cm.a.227.4 16
56.27 even 2 inner 3528.1.co.a.587.6 16
56.51 odd 6 3528.1.cm.a.227.1 16
63.5 even 6 3528.1.u.a.2579.3 16
63.23 odd 6 3528.1.u.a.2579.2 16
63.32 odd 6 3528.1.cm.a.3155.8 16
63.41 even 6 inner 3528.1.co.a.2939.7 yes 16
63.59 even 6 3528.1.cm.a.3155.5 16
72.59 even 6 inner 3528.1.co.a.2939.6 yes 16
504.59 odd 6 3528.1.cm.a.3155.5 16
504.131 odd 6 3528.1.u.a.2579.3 16
504.275 even 6 3528.1.u.a.2579.2 16
504.347 even 6 3528.1.cm.a.3155.8 16
504.419 odd 6 inner 3528.1.co.a.2939.7 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3528.1.u.a.803.2 16 7.3 odd 6
3528.1.u.a.803.2 16 56.3 even 6
3528.1.u.a.803.3 16 7.4 even 3
3528.1.u.a.803.3 16 56.11 odd 6
3528.1.u.a.2579.2 16 63.23 odd 6
3528.1.u.a.2579.2 16 504.275 even 6
3528.1.u.a.2579.3 16 63.5 even 6
3528.1.u.a.2579.3 16 504.131 odd 6
3528.1.cm.a.227.1 16 7.2 even 3
3528.1.cm.a.227.1 16 56.51 odd 6
3528.1.cm.a.227.4 16 7.5 odd 6
3528.1.cm.a.227.4 16 56.19 even 6
3528.1.cm.a.3155.5 16 63.59 even 6
3528.1.cm.a.3155.5 16 504.59 odd 6
3528.1.cm.a.3155.8 16 63.32 odd 6
3528.1.cm.a.3155.8 16 504.347 even 6
3528.1.co.a.587.6 16 7.6 odd 2 inner
3528.1.co.a.587.6 16 56.27 even 2 inner
3528.1.co.a.587.7 yes 16 1.1 even 1 trivial
3528.1.co.a.587.7 yes 16 8.3 odd 2 CM
3528.1.co.a.2939.6 yes 16 9.5 odd 6 inner
3528.1.co.a.2939.6 yes 16 72.59 even 6 inner
3528.1.co.a.2939.7 yes 16 63.41 even 6 inner
3528.1.co.a.2939.7 yes 16 504.419 odd 6 inner