Properties

Label 1620.1.w.a.863.2
Level $1620$
Weight $1$
Character 1620.863
Analytic conductor $0.808$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,1,Mod(107,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.107");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1620.w (of order \(12\), degree \(4\), 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: \(0.808485320465\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

Embedding invariants

Embedding label 863.2
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1620.863
Dual form 1620.1.w.a.107.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.258819 - 0.965926i) q^{2} +(-0.866025 - 0.500000i) q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.707107 + 0.707107i) q^{8} +O(q^{10})\) \(q+(0.258819 - 0.965926i) q^{2} +(-0.866025 - 0.500000i) q^{4} +(-0.707107 - 0.707107i) q^{5} +(-0.707107 + 0.707107i) q^{8} +(-0.866025 + 0.500000i) q^{10} +(-1.86603 + 0.500000i) q^{13} +(0.500000 + 0.866025i) q^{16} +(-1.22474 - 1.22474i) q^{17} +(0.258819 + 0.965926i) q^{20} +1.00000i q^{25} +1.93185i q^{26} +(-0.258819 - 0.448288i) q^{29} +(0.965926 - 0.258819i) q^{32} +(-1.50000 + 0.866025i) q^{34} +(-0.366025 + 0.366025i) q^{37} +1.00000 q^{40} +(-1.22474 - 0.707107i) q^{41} +(0.866025 + 0.500000i) q^{49} +(0.965926 + 0.258819i) q^{50} +(1.86603 + 0.500000i) q^{52} +(-0.500000 + 0.133975i) q^{58} +(-0.866025 - 1.50000i) q^{61} -1.00000i q^{64} +(1.67303 + 0.965926i) q^{65} +(0.448288 + 1.67303i) q^{68} +(0.366025 + 0.366025i) q^{73} +(0.258819 + 0.448288i) q^{74} +(0.258819 - 0.965926i) q^{80} +(-1.00000 + 1.00000i) q^{82} +1.73205i q^{85} +1.93185 q^{89} +(-1.36603 - 0.366025i) q^{97} +(0.707107 - 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} + 4 q^{16} - 12 q^{34} + 4 q^{37} + 8 q^{40} + 8 q^{52} - 4 q^{58} - 4 q^{73} - 8 q^{82} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.258819 0.965926i 0.258819 0.965926i
\(3\) 0 0
\(4\) −0.866025 0.500000i −0.866025 0.500000i
\(5\) −0.707107 0.707107i −0.707107 0.707107i
\(6\) 0 0
\(7\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(8\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(9\) 0 0
\(10\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −1.86603 + 0.500000i −1.86603 + 0.500000i −0.866025 + 0.500000i \(0.833333\pi\)
−1.00000 \(1.00000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(17\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(24\) 0 0
\(25\) 1.00000i 1.00000i
\(26\) 1.93185i 1.93185i
\(27\) 0 0
\(28\) 0 0
\(29\) −0.258819 0.448288i −0.258819 0.448288i 0.707107 0.707107i \(-0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0.965926 0.258819i 0.965926 0.258819i
\(33\) 0 0
\(34\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(35\) 0 0
\(36\) 0 0
\(37\) −0.366025 + 0.366025i −0.366025 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 1.00000
\(41\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(42\) 0 0
\(43\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(48\) 0 0
\(49\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(50\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(51\) 0 0
\(52\) 1.86603 + 0.500000i 1.86603 + 0.500000i
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.500000 + 0.133975i −0.500000 + 0.133975i
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) 1.67303 + 0.965926i 1.67303 + 0.965926i
\(66\) 0 0
\(67\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(68\) 0.448288 + 1.67303i 0.448288 + 1.67303i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0.366025 + 0.366025i 0.366025 + 0.366025i 0.866025 0.500000i \(-0.166667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0.258819 + 0.448288i 0.258819 + 0.448288i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0.258819 0.965926i 0.258819 0.965926i
\(81\) 0 0
\(82\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(83\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(84\) 0 0
\(85\) 1.73205i 1.73205i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.93185 1.93185 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.36603 0.366025i −1.36603 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(98\) 0.707107 0.707107i 0.707107 0.707107i
\(99\) 0 0
\(100\) 0.500000 0.866025i 0.500000 0.866025i
\(101\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(102\) 0 0
\(103\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(104\) 0.965926 1.67303i 0.965926 1.67303i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.258819 0.965926i −0.258819 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.517638i 0.517638i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 0.866025i 0.500000 0.866025i
\(122\) −1.67303 + 0.448288i −1.67303 + 0.448288i
\(123\) 0 0
\(124\) 0 0
\(125\) 0.707107 0.707107i 0.707107 0.707107i
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) −0.965926 0.258819i −0.965926 0.258819i
\(129\) 0 0
\(130\) 1.36603 1.36603i 1.36603 1.36603i
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 1.73205 1.73205
\(137\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(138\) 0 0
\(139\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −0.133975 + 0.500000i −0.133975 + 0.500000i
\(146\) 0.448288 0.258819i 0.448288 0.258819i
\(147\) 0 0
\(148\) 0.500000 0.133975i 0.500000 0.133975i
\(149\) −0.965926 + 1.67303i −0.965926 + 1.67303i −0.258819 + 0.965926i \(0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) 0 0
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.500000 1.86603i −0.500000 1.86603i −0.500000 0.866025i \(-0.666667\pi\)
1.00000i \(-0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −0.866025 0.500000i −0.866025 0.500000i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(168\) 0 0
\(169\) 2.36603 1.36603i 2.36603 1.36603i
\(170\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.500000 1.86603i 0.500000 1.86603i
\(179\) 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
\(185\) 0.517638 0.517638
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) 1.86603 0.500000i 1.86603 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000 \(0\)
\(194\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(195\) 0 0
\(196\) −0.500000 0.866025i −0.500000 0.866025i
\(197\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −0.707107 0.707107i −0.707107 0.707107i
\(201\) 0 0
\(202\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(206\) 0 0
\(207\) 0 0
\(208\) −1.36603 1.36603i −1.36603 1.36603i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −1.67303 0.448288i −1.67303 0.448288i
\(219\) 0 0
\(220\) 0 0
\(221\) 2.89778 + 1.67303i 2.89778 + 1.67303i
\(222\) 0 0
\(223\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1.00000 −1.00000
\(227\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(228\) 0 0
\(229\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.500000 + 0.133975i 0.500000 + 0.133975i
\(233\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(240\) 0 0
\(241\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(242\) −0.707107 0.707107i −0.707107 0.707107i
\(243\) 0 0
\(244\) 1.73205i 1.73205i
\(245\) −0.258819 0.965926i −0.258819 0.965926i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.500000 0.866025i −0.500000 0.866025i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.965926 1.67303i −0.965926 1.67303i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.517638 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0.448288 1.67303i 0.448288 1.67303i
\(273\) 0 0
\(274\) −0.866025 0.500000i −0.866025 0.500000i
\(275\) 0 0
\(276\) 0 0
\(277\) 1.36603 + 0.366025i 1.36603 + 0.366025i 0.866025 0.500000i \(-0.166667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.448288 + 0.258819i −0.448288 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(282\) 0 0
\(283\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.00000i 2.00000i
\(290\) 0.448288 + 0.258819i 0.448288 + 0.258819i
\(291\) 0 0
\(292\) −0.133975 0.500000i −0.133975 0.500000i
\(293\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.517638i 0.517638i
\(297\) 0 0
\(298\) 1.36603 + 1.36603i 1.36603 + 1.36603i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.448288 + 1.67303i −0.448288 + 1.67303i
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) −0.133975 + 0.500000i −0.133975 + 0.500000i 0.866025 + 0.500000i \(0.166667\pi\)
−1.00000 \(\pi\)
\(314\) −1.93185 −1.93185
\(315\) 0 0
\(316\) 0 0
\(317\) −0.448288 + 1.67303i −0.448288 + 1.67303i 0.258819 + 0.965926i \(0.416667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −0.500000 1.86603i −0.500000 1.86603i
\(326\) 0 0
\(327\) 0 0
\(328\) 1.36603 0.366025i 1.36603 0.366025i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.366025 1.36603i −0.366025 1.36603i −0.866025 0.500000i \(-0.833333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(338\) −0.707107 2.63896i −0.707107 2.63896i
\(339\) 0 0
\(340\) 0.866025 1.50000i 0.866025 1.50000i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(347\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(348\) 0 0
\(349\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.67303 0.965926i −1.67303 0.965926i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.517638i 0.517638i
\(366\) 0 0
\(367\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0.133975 0.500000i 0.133975 0.500000i
\(371\) 0 0
\(372\) 0 0
\(373\) −1.36603 + 0.366025i −1.36603 + 0.366025i −0.866025 0.500000i \(-0.833333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.93185i 1.93185i
\(387\) 0 0
\(388\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(389\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(393\) 0 0
\(394\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(395\) 0 0
\(396\) 0 0
\(397\) −1.36603 + 1.36603i −1.36603 + 1.36603i −0.500000 + 0.866025i \(0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(401\) 0.448288 + 0.258819i 0.448288 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.41421 1.41421
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(410\) 1.41421 1.41421
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −1.67303 + 0.965926i −1.67303 + 0.965926i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.22474 1.22474i 1.22474 1.22474i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −0.366025 0.366025i −0.366025 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.36603 2.36603i 2.36603 2.36603i
\(443\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(444\) 0 0
\(445\) −1.36603 1.36603i −1.36603 1.36603i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.500000 0.133975i −0.500000 0.133975i 1.00000i \(-0.5\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0.707107 0.707107i 0.707107 0.707107i
\(459\) 0 0
\(460\) 0 0
\(461\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(462\) 0 0
\(463\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(464\) 0.258819 0.448288i 0.258819 0.448288i
\(465\) 0 0
\(466\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 0.500000 0.866025i 0.500000 0.866025i
\(482\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(483\) 0 0
\(484\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(485\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(489\) 0 0
\(490\) −1.00000 −1.00000
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) −0.232051 + 0.866025i −0.232051 + 0.866025i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(500\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(513\) 0 0
\(514\) 1.00000i 1.00000i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −1.86603 + 0.500000i −1.86603 + 0.500000i
\(521\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.63896 + 0.707107i 2.63896 + 0.707107i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.133975 0.500000i 0.133975 0.500000i
\(539\) 0 0
\(540\) 0 0
\(541\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.50000 0.866025i −1.50000 0.866025i
\(545\) −1.22474 + 1.22474i −1.22474 + 1.22474i
\(546\) 0 0
\(547\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(548\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.707107 1.22474i 0.707107 1.22474i
\(555\) 0 0
\(556\) 0 0
\(557\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.133975 + 0.500000i 0.133975 + 0.500000i
\(563\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(564\) 0 0
\(565\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.965926 + 1.67303i 0.965926 + 1.67303i 0.707107 + 0.707107i \(0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.36603 1.36603i 1.36603 1.36603i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(578\) 1.93185 + 0.517638i 1.93185 + 0.517638i
\(579\) 0 0
\(580\) 0.366025 0.366025i 0.366025 0.366025i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.517638 −0.517638
\(585\) 0 0
\(586\) 1.00000 1.00000
\(587\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.500000 0.133975i −0.500000 0.133975i
\(593\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.67303 0.965926i 1.67303 0.965926i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(606\) 0 0
\(607\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.67303 0.448288i 1.67303 0.448288i 0.707107 0.707107i \(-0.250000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(618\) 0 0
\(619\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −1.00000
\(626\) 0.448288 + 0.258819i 0.448288 + 0.258819i
\(627\) 0 0
\(628\) −0.500000 + 1.86603i −0.500000 + 1.86603i
\(629\) 0.896575 0.896575
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(635\) 0 0
\(636\) 0 0
\(637\) −1.86603 0.500000i −1.86603 0.500000i
\(638\) 0 0
\(639\) 0 0
\(640\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(641\) −1.67303 + 0.965926i −1.67303 + 0.965926i −0.707107 + 0.707107i \(0.750000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(642\) 0 0
\(643\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.93185 −1.93185
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.41421i 1.41421i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 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 + 1.86603i −0.500000 + 1.86603i 1.00000i \(0.5\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) −1.41421 −1.41421
\(675\) 0 0
\(676\) −2.73205 −2.73205
\(677\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1.22474 1.22474i −1.22474 1.22474i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 0 0
\(685\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.633975 + 2.36603i 0.633975 + 2.36603i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.36603 + 1.36603i −1.36603 + 1.36603i
\(713\) 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
\(722\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(723\) 0 0
\(724\) 0 0
\(725\) 0.448288 0.258819i 0.448288 0.258819i
\(726\) 0 0
\(727\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −0.500000 0.133975i −0.500000 0.133975i
\(731\) 0 0
\(732\) 0 0
\(733\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −0.448288 0.258819i −0.448288 0.258819i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(744\) 0 0
\(745\) 1.86603 0.500000i 1.86603 0.500000i
\(746\) 1.41421i 1.41421i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0.866025 0.500000i 0.866025 0.500000i
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.448288 0.258819i −0.448288 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.86603 0.500000i −1.86603 0.500000i
\(773\) 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.22474 0.707107i 1.22474 0.707107i
\(777\) 0 0
\(778\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000i 1.00000i
\(785\) −0.965926 + 1.67303i −0.965926 + 1.67303i
\(786\) 0 0
\(787\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(788\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.36603 + 2.36603i 2.36603 + 2.36603i
\(794\) 0.965926 + 1.67303i 0.965926 + 1.67303i
\(795\) 0 0
\(796\) 0 0
\(797\) −0.965926 + 0.258819i −0.965926 + 0.258819i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(801\) 0 0
\(802\) 0.366025 0.366025i 0.366025 0.366025i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.366025 1.36603i 0.366025 1.36603i
\(809\) −1.93185 −1.93185 −0.965926 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.22474 1.22474i 1.22474 1.22474i
\(819\) 0 0
\(820\) 0.366025 1.36603i 0.366025 1.36603i
\(821\) 1.67303 0.965926i 1.67303 0.965926i 0.707107 0.707107i \(-0.250000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(822\) 0 0
\(823\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.500000 + 1.86603i 0.500000 + 1.86603i
\(833\) −0.448288 1.67303i −0.448288 1.67303i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 0 0
\(841\) 0.366025 0.633975i 0.366025 0.633975i
\(842\) 0.965926 0.258819i 0.965926 0.258819i
\(843\) 0 0
\(844\) 0 0
\(845\) −2.63896 0.707107i −2.63896 0.707107i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −0.866025 1.50000i −0.866025 1.50000i
\(851\) 0 0
\(852\) 0 0
\(853\) 0.366025 1.36603i 0.366025 1.36603i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.448288 1.67303i 0.448288 1.67303i −0.258819 0.965926i \(-0.583333\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(858\) 0 0
\(859\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0 0
\(865\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(866\) −0.448288 + 0.258819i −0.448288 + 0.258819i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.133975 0.500000i −0.133975 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(882\) 0 0
\(883\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) −1.67303 2.89778i −1.67303 2.89778i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.67303 + 0.965926i −1.67303 + 0.965926i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.258819 + 0.448288i −0.258819 + 0.448288i
\(915\) 0 0
\(916\) −0.500000 0.866025i −0.500000 0.866025i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −0.366025 1.36603i −0.366025 1.36603i
\(923\) 0 0
\(924\) 0 0
\(925\) −0.366025 0.366025i −0.366025 0.366025i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.366025 0.366025i −0.366025 0.366025i
\(929\) −0.965926 1.67303i −0.965926 1.67303i −0.707107 0.707107i \(-0.750000\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.67303 0.448288i 1.67303 0.448288i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.366025 0.366025i 0.366025 0.366025i −0.500000 0.866025i \(-0.666667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.67303 0.965926i −1.67303 0.965926i −0.965926 0.258819i \(-0.916667\pi\)
−0.707107 0.707107i \(-0.750000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(948\) 0 0
\(949\) −0.866025 0.500000i −0.866025 0.500000i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(954\) 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
\(962\) −0.707107 0.707107i −0.707107 0.707107i
\(963\) 0 0
\(964\) 1.00000i 1.00000i
\(965\) −1.67303 0.965926i −1.67303 0.965926i
\(966\) 0 0
\(967\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(968\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(969\) 0 0
\(970\) 1.36603 0.366025i 1.36603 0.366025i
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0.866025 1.50000i 0.866025 1.50000i
\(977\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(984\) 0 0
\(985\) 1.00000i 1.00000i
\(986\) 0.776457 + 0.448288i 0.776457 + 0.448288i
\(987\) 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 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.500000 + 0.133975i 0.500000 + 0.133975i 0.500000 0.866025i \(-0.333333\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 1620.1.w.a.863.2 8
3.2 odd 2 inner 1620.1.w.a.863.1 8
4.3 odd 2 CM 1620.1.w.a.863.2 8
5.2 odd 4 1620.1.w.c.1187.2 8
9.2 odd 6 1620.1.w.c.1403.2 8
9.4 even 3 1620.1.m.a.323.3 yes 8
9.5 odd 6 1620.1.m.a.323.2 8
9.7 even 3 1620.1.w.c.1403.1 8
12.11 even 2 inner 1620.1.w.a.863.1 8
15.2 even 4 1620.1.w.c.1187.1 8
20.7 even 4 1620.1.w.c.1187.2 8
36.7 odd 6 1620.1.w.c.1403.1 8
36.11 even 6 1620.1.w.c.1403.2 8
36.23 even 6 1620.1.m.a.323.2 8
36.31 odd 6 1620.1.m.a.323.3 yes 8
45.2 even 12 inner 1620.1.w.a.107.2 8
45.7 odd 12 inner 1620.1.w.a.107.1 8
45.22 odd 12 1620.1.m.a.647.2 yes 8
45.32 even 12 1620.1.m.a.647.3 yes 8
60.47 odd 4 1620.1.w.c.1187.1 8
180.7 even 12 inner 1620.1.w.a.107.1 8
180.47 odd 12 inner 1620.1.w.a.107.2 8
180.67 even 12 1620.1.m.a.647.2 yes 8
180.167 odd 12 1620.1.m.a.647.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.1.m.a.323.2 8 9.5 odd 6
1620.1.m.a.323.2 8 36.23 even 6
1620.1.m.a.323.3 yes 8 9.4 even 3
1620.1.m.a.323.3 yes 8 36.31 odd 6
1620.1.m.a.647.2 yes 8 45.22 odd 12
1620.1.m.a.647.2 yes 8 180.67 even 12
1620.1.m.a.647.3 yes 8 45.32 even 12
1620.1.m.a.647.3 yes 8 180.167 odd 12
1620.1.w.a.107.1 8 45.7 odd 12 inner
1620.1.w.a.107.1 8 180.7 even 12 inner
1620.1.w.a.107.2 8 45.2 even 12 inner
1620.1.w.a.107.2 8 180.47 odd 12 inner
1620.1.w.a.863.1 8 3.2 odd 2 inner
1620.1.w.a.863.1 8 12.11 even 2 inner
1620.1.w.a.863.2 8 1.1 even 1 trivial
1620.1.w.a.863.2 8 4.3 odd 2 CM
1620.1.w.c.1187.1 8 15.2 even 4
1620.1.w.c.1187.1 8 60.47 odd 4
1620.1.w.c.1187.2 8 5.2 odd 4
1620.1.w.c.1187.2 8 20.7 even 4
1620.1.w.c.1403.1 8 9.7 even 3
1620.1.w.c.1403.1 8 36.7 odd 6
1620.1.w.c.1403.2 8 9.2 odd 6
1620.1.w.c.1403.2 8 36.11 even 6