Properties

Label 735.1.o.c.569.2
Level $735$
Weight $1$
Character 735.569
Analytic conductor $0.367$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -15
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,1,Mod(569,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.569");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 735.o (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: \(0.366812784285\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.15435.1
Artin image: $C_6\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

Embedding invariants

Embedding label 569.2
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 735.569
Dual form 735.1.o.c.704.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 1.22474i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} -1.41421 q^{6} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.707107 - 1.22474i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} -1.41421 q^{6} +(-0.500000 + 0.866025i) q^{9} +(-0.707107 - 1.22474i) q^{10} +(-0.500000 + 0.866025i) q^{12} -1.00000 q^{15} +(0.500000 - 0.866025i) q^{16} +(0.707107 + 1.22474i) q^{18} +(-0.707107 + 1.22474i) q^{19} -1.00000 q^{20} +(-0.707107 + 1.22474i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +(-0.707107 + 1.22474i) q^{30} +(0.707107 + 1.22474i) q^{31} +(-0.707107 - 1.22474i) q^{32} +1.00000 q^{36} +(1.00000 + 1.73205i) q^{38} +(0.500000 + 0.866025i) q^{45} +(1.00000 + 1.73205i) q^{46} -1.00000 q^{48} -1.41421 q^{50} +(-0.707107 - 1.22474i) q^{53} +(0.707107 - 1.22474i) q^{54} +1.41421 q^{57} +(0.500000 + 0.866025i) q^{60} +(0.707107 - 1.22474i) q^{61} +2.00000 q^{62} -1.00000 q^{64} +1.41421 q^{69} +(-0.500000 + 0.866025i) q^{75} +1.41421 q^{76} +(-0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +1.41421 q^{90} +1.41421 q^{92} +(0.707107 - 1.22474i) q^{93} +(0.707107 + 1.22474i) q^{95} +(-0.707107 + 1.22474i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{9} - 2 q^{12} - 4 q^{15} + 2 q^{16} - 4 q^{20} - 2 q^{25} + 4 q^{27} + 4 q^{36} + 4 q^{38} + 2 q^{45} + 4 q^{46} - 4 q^{48} + 2 q^{60} + 8 q^{62} - 4 q^{64} - 2 q^{75} - 2 q^{80} - 2 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(3\) −0.500000 0.866025i −0.500000 0.866025i
\(4\) −0.500000 0.866025i −0.500000 0.866025i
\(5\) 0.500000 0.866025i 0.500000 0.866025i
\(6\) −1.41421 −1.41421
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) −0.707107 1.22474i −0.707107 1.22474i
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −1.00000 −1.00000
\(16\) 0.500000 0.866025i 0.500000 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(19\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(20\) −1.00000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.500000 0.866025i
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(31\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(32\) −0.707107 1.22474i −0.707107 1.22474i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(46\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) −1.00000 −1.00000
\(49\) 0 0
\(50\) −1.41421 −1.41421
\(51\) 0 0
\(52\) 0 0
\(53\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(54\) 0.707107 1.22474i 0.707107 1.22474i
\(55\) 0 0
\(56\) 0 0
\(57\) 1.41421 1.41421
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(61\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(62\) 2.00000 2.00000
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 1.41421 1.41421
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(76\) 1.41421 1.41421
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) −0.500000 0.866025i −0.500000 0.866025i
\(81\) −0.500000 0.866025i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(90\) 1.41421 1.41421
\(91\) 0 0
\(92\) 1.41421 1.41421
\(93\) 0.707107 1.22474i 0.707107 1.22474i
\(94\) 0 0
\(95\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(96\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −2.00000
\(107\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(108\) −0.500000 0.866025i −0.500000 0.866025i
\(109\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 1.00000 1.73205i 1.00000 1.73205i
\(115\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(122\) −1.00000 1.73205i −1.00000 1.73205i
\(123\) 0 0
\(124\) 0.707107 1.22474i 0.707107 1.22474i
\(125\) −1.00000 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.500000 0.866025i 0.500000 0.866025i
\(136\) 0 0
\(137\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(138\) 1.00000 1.73205i 1.00000 1.73205i
\(139\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(151\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.41421 1.41421
\(156\) 0 0
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 0 0
\(159\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(160\) −1.41421 −1.41421
\(161\) 0 0
\(162\) −1.41421 −1.41421
\(163\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) −0.707107 1.22474i −0.707107 1.22474i
\(172\) 0 0
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0.500000 0.866025i 0.500000 0.866025i
\(181\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) 0 0
\(183\) −1.41421 −1.41421
\(184\) 0 0
\(185\) 0 0
\(186\) −1.00000 1.73205i −1.00000 1.73205i
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 2.00000 2.00000
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.707107 1.22474i −0.707107 1.22474i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(212\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(213\) 0 0
\(214\) −1.00000 1.73205i −1.00000 1.73205i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.82843 2.82843
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 1.00000 1.00000
\(226\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(227\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) −0.707107 1.22474i −0.707107 1.22474i
\(229\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(230\) 2.00000 2.00000
\(231\) 0 0
\(232\) 0 0
\(233\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−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 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(241\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(242\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(243\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(244\) −1.41421 −1.41421
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.500000 0.866025i
\(257\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(264\) 0 0
\(265\) −1.41421 −1.41421
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) −0.707107 1.22474i −0.707107 1.22474i
\(271\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 2.00000 2.00000
\(275\) 0 0
\(276\) −0.707107 1.22474i −0.707107 1.22474i
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(279\) −1.41421 −1.41421
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 0 0
\(285\) 0.707107 1.22474i 0.707107 1.22474i
\(286\) 0 0
\(287\) 0 0
\(288\) 1.41421 1.41421
\(289\) 0.500000 0.866025i 0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 1.00000 1.00000
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(305\) −0.707107 1.22474i −0.707107 1.22474i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.00000 1.73205i 1.00000 1.73205i
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(318\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(319\) 0 0
\(320\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(321\) −1.41421 −1.41421
\(322\) 0 0
\(323\) 0 0
\(324\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(325\) 0 0
\(326\) 0 0
\(327\) 1.00000 1.73205i 1.00000 1.73205i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −1.41421 + 2.44949i −1.41421 + 2.44949i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0.707107 1.22474i 0.707107 1.22474i
\(339\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(340\) 0 0
\(341\) 0 0
\(342\) −2.00000 −2.00000
\(343\) 0 0
\(344\) 0 0
\(345\) 0.707107 1.22474i 0.707107 1.22474i
\(346\) 0 0
\(347\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(348\) 0 0
\(349\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) −0.500000 0.866025i −0.500000 0.866025i
\(362\) 1.00000 1.73205i 1.00000 1.73205i
\(363\) 1.00000 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(367\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.41421 −1.41421
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0.707107 1.22474i 0.707107 1.22474i
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) −2.00000 −2.00000
\(399\) 0 0
\(400\) −1.00000 −1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(410\) 0 0
\(411\) 0.707107 1.22474i 0.707107 1.22474i
\(412\) 0 0
\(413\) 0 0
\(414\) −2.00000 −2.00000
\(415\) 0 0
\(416\) 0 0
\(417\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) −1.41421 + 2.44949i −1.41421 + 2.44949i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.41421 −1.41421
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 0.500000 0.866025i 0.500000 0.866025i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.00000 1.73205i 1.00000 1.73205i
\(437\) −1.00000 1.73205i −1.00000 1.73205i
\(438\) 0 0
\(439\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0.707107 1.22474i 0.707107 1.22474i
\(451\) 0 0
\(452\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(453\) 0 0
\(454\) 2.82843 2.82843
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) −1.00000 1.73205i −1.00000 1.73205i
\(459\) 0 0
\(460\) 0.707107 1.22474i 0.707107 1.22474i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) −0.707107 1.22474i −0.707107 1.22474i
\(466\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(467\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.41421 1.41421
\(476\) 0 0
\(477\) 1.41421 1.41421
\(478\) 0 0
\(479\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(481\) 0 0
\(482\) −2.00000 −2.00000
\(483\) 0 0
\(484\) 1.00000 1.00000
\(485\) 0 0
\(486\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(487\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.41421 1.41421
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(500\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(501\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.500000 0.866025i −0.500000 0.866025i
\(508\) 0 0
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.41421 −1.41421
\(513\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(514\) 1.41421 + 2.44949i 1.41421 + 2.44949i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.00000 2.00000
\(527\) 0 0
\(528\) 0 0
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.707107 1.22474i −0.707107 1.22474i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −1.00000 −1.00000
\(541\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(542\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(543\) −0.707107 1.22474i −0.707107 1.22474i
\(544\) 0 0
\(545\) 2.00000 2.00000
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0.707107 1.22474i 0.707107 1.22474i
\(549\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(557\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(558\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) −1.00000 1.73205i −1.00000 1.73205i
\(571\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.41421 1.41421
\(576\) 0.500000 0.866025i 0.500000 0.866025i
\(577\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(578\) −0.707107 1.22474i −0.707107 1.22474i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 1.41421 2.44949i 1.41421 2.44949i
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −2.00000 −2.00000
\(590\) 0 0
\(591\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(592\) 0 0
\(593\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 2.00000 2.00000
\(609\) 0 0
\(610\) −2.00000 −2.00000
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(618\) 0 0
\(619\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(620\) −0.707107 1.22474i −0.707107 1.22474i
\(621\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(634\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(635\) 0 0
\(636\) 1.41421 1.41421
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(654\) −1.41421 2.44949i −1.41421 2.44949i
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(662\) −1.41421 2.44949i −1.41421 2.44949i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −0.500000 0.866025i −0.500000 0.866025i
\(676\) −0.500000 0.866025i −0.500000 0.866025i
\(677\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 2.00000 2.00000
\(679\) 0 0
\(680\) 0 0
\(681\) 1.00000 1.73205i 1.00000 1.73205i
\(682\) 0 0
\(683\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(684\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(685\) 1.41421 1.41421
\(686\) 0 0
\(687\) −1.41421 −1.41421
\(688\) 0 0
\(689\) 0 0
\(690\) −1.00000 1.73205i −1.00000 1.73205i
\(691\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −2.00000 −2.00000
\(695\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(696\) 0 0
\(697\) 0 0
\(698\) 1.00000 1.73205i 1.00000 1.73205i
\(699\) 1.41421 1.41421
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −2.82843 −2.82843
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.00000 −2.00000
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 1.00000 1.00000
\(721\) 0 0
\(722\) −1.41421 −1.41421
\(723\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(724\) −0.707107 1.22474i −0.707107 1.22474i
\(725\) 0 0
\(726\) 0.707107 1.22474i 0.707107 1.22474i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(733\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 2.00000 2.00000
\(737\) 0 0
\(738\) 0 0
\(739\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 1.41421 1.41421
\(751\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(769\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(770\) 0 0
\(771\) 2.00000 2.00000
\(772\) 0 0
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 0.707107 1.22474i 0.707107 1.22474i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(789\) 0.707107 1.22474i 0.707107 1.22474i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(796\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)