Properties

Label 1045.1.k.a.417.1
Level $1045$
Weight $1$
Character 1045.417
Analytic conductor $0.522$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
RM discriminant 209
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1045,1,Mod(208,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.208");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1045.k (of order \(4\), 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: \(0.521522938201\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.26125.1

Embedding invariants

Embedding label 417.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1045.417
Dual form 1045.1.k.a.208.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{4} -1.00000i q^{5} -1.00000i q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{4} -1.00000i q^{5} -1.00000i q^{9} +(-1.00000 + 1.00000i) q^{10} +1.00000 q^{11} +(1.00000 - 1.00000i) q^{13} +1.00000 q^{16} +(-1.00000 + 1.00000i) q^{18} +1.00000i q^{19} +1.00000 q^{20} +(-1.00000 - 1.00000i) q^{22} +(-1.00000 + 1.00000i) q^{23} -1.00000 q^{25} -2.00000 q^{26} -2.00000i q^{29} +(-1.00000 - 1.00000i) q^{32} +1.00000 q^{36} +(1.00000 - 1.00000i) q^{38} +1.00000i q^{44} -1.00000 q^{45} +2.00000 q^{46} +(-1.00000 - 1.00000i) q^{47} +1.00000i q^{49} +(1.00000 + 1.00000i) q^{50} +(1.00000 + 1.00000i) q^{52} -1.00000i q^{55} +(-2.00000 + 2.00000i) q^{58} +1.00000i q^{64} +(-1.00000 - 1.00000i) q^{65} -1.00000 q^{76} -1.00000i q^{80} -1.00000 q^{81} +(1.00000 + 1.00000i) q^{90} +(-1.00000 - 1.00000i) q^{92} +2.00000i q^{94} +1.00000 q^{95} +(1.00000 - 1.00000i) q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{10} + 2 q^{11} + 2 q^{13} + 2 q^{16} - 2 q^{18} + 2 q^{20} - 2 q^{22} - 2 q^{23} - 2 q^{25} - 4 q^{26} - 2 q^{32} + 2 q^{36} + 2 q^{38} - 2 q^{45} + 4 q^{46} - 2 q^{47} + 2 q^{50} + 2 q^{52} - 4 q^{58} - 2 q^{65} - 2 q^{76} - 2 q^{81} + 2 q^{90} - 2 q^{92} + 2 q^{95} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(3\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 1.00000i 1.00000i
\(5\) 1.00000i 1.00000i
\(6\) 0 0
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0 0
\(9\) 1.00000i 1.00000i
\(10\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(11\) 1.00000 1.00000
\(12\) 0 0
\(13\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(19\) 1.00000i 1.00000i
\(20\) 1.00000 1.00000
\(21\) 0 0
\(22\) −1.00000 1.00000i −1.00000 1.00000i
\(23\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −1.00000 −1.00000
\(26\) −2.00000 −2.00000
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 1.00000i −1.00000 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 1.00000 1.00000i 1.00000 1.00000i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 1.00000i 1.00000i
\(45\) −1.00000 −1.00000
\(46\) 2.00000 2.00000
\(47\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(51\) 0 0
\(52\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 + 2.00000i −2.00000 + 2.00000i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000i 1.00000i
\(65\) −1.00000 1.00000i −1.00000 1.00000i
\(66\) 0 0
\(67\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.00000 −1.00000
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.00000i 1.00000i
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(91\) 0 0
\(92\) −1.00000 1.00000i −1.00000 1.00000i
\(93\) 0 0
\(94\) 2.00000i 2.00000i
\(95\) 1.00000 1.00000
\(96\) 0 0
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) 1.00000 1.00000i 1.00000 1.00000i
\(99\) 1.00000i 1.00000i
\(100\) 1.00000i 1.00000i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(114\) 0 0
\(115\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(116\) 2.00000 2.00000
\(117\) −1.00000 1.00000i −1.00000 1.00000i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000i 1.00000i
\(126\) 0 0
\(127\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 2.00000i 2.00000i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.00000 1.00000i 1.00000 1.00000i
\(144\) 1.00000i 1.00000i
\(145\) −2.00000 −2.00000
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(161\) 0 0
\(162\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(163\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000i 1.00000i
\(170\) 0 0
\(171\) 1.00000 1.00000
\(172\) 0 0
\(173\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 1.00000
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 1.00000i 1.00000i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 1.00000 1.00000i 1.00000 1.00000i
\(189\) 0 0
\(190\) −1.00000 1.00000i −1.00000 1.00000i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.00000 −1.00000
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(208\) 1.00000 1.00000i 1.00000 1.00000i
\(209\) 1.00000i 1.00000i
\(210\) 0 0
\(211\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 2.00000i 2.00000i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 2.00000i 2.00000 2.00000i
\(219\) 0 0
\(220\) 1.00000 1.00000
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 1.00000i 1.00000i
\(226\) 0 0
\(227\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(230\) 2.00000i 2.00000i
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 2.00000i 2.00000i
\(235\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(242\) −1.00000 1.00000i −1.00000 1.00000i
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 1.00000
\(246\) 0 0
\(247\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(248\) 0 0
\(249\) 0 0
\(250\) 1.00000 1.00000i 1.00000 1.00000i
\(251\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(254\) 2.00000i 2.00000i
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.00000 1.00000i 1.00000 1.00000i
\(261\) −2.00000 −2.00000
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 2.00000i 2.00000i
\(275\) −1.00000 −1.00000
\(276\) 0 0
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.00000 −2.00000
\(287\) 0 0
\(288\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(289\) 1.00000i 1.00000i
\(290\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(291\) 0 0
\(292\) 0 0
\(293\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.00000i 2.00000i
\(300\) 0 0
\(301\) 0 0
\(302\) −2.00000 2.00000i −2.00000 2.00000i
\(303\) 0 0
\(304\) 1.00000i 1.00000i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(314\) 2.00000i 2.00000i
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 2.00000i 2.00000i
\(320\) 1.00000 1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000i 1.00000i
\(325\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(326\) −2.00000 −2.00000
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 2.00000i 2.00000i
\(335\) 0 0
\(336\) 0 0
\(337\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(338\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −1.00000 1.00000i −1.00000 1.00000i
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −2.00000 −2.00000
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 1.00000i −1.00000 1.00000i
\(353\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(368\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 2.00000i −2.00000 2.00000i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 1.00000i 1.00000i
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −2.00000
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 1.00000 1.00000
\(397\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000i 1.00000i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 2.00000i 2.00000i
\(415\) 0 0
\(416\) −2.00000 −2.00000
\(417\) 0 0
\(418\) 1.00000 1.00000i 1.00000 1.00000i
\(419\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(423\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −2.00000
\(437\) −1.00000 1.00000i −1.00000 1.00000i
\(438\) 0 0
\(439\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0 0
\(443\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 1.00000 1.00000i 1.00000 1.00000i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 2.00000i 2.00000i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) −2.00000 + 2.00000i −2.00000 + 2.00000i
\(459\) 0 0
\(460\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(464\) 2.00000i 2.00000i
\(465\) 0 0
\(466\) 0 0
\(467\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(468\) 1.00000 1.00000i 1.00000 1.00000i
\(469\) 0 0
\(470\) 2.00000 2.00000
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000i 1.00000i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(483\) 0 0
\(484\) 1.00000i 1.00000i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −1.00000 1.00000i −1.00000 1.00000i
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 2.00000i 2.00000i
\(495\) −1.00000 −1.00000
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(500\) −1.00000 −1.00000
\(501\) 0 0
\(502\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 2.00000
\(507\) 0 0
\(508\) 1.00000 1.00000i 1.00000 1.00000i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 1.00000i −1.00000 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.00000 1.00000i −1.00000 1.00000i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(523\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 1.00000 1.00000i 1.00000 1.00000i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.00000i 1.00000i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.00000 2.00000
\(546\) 0 0
\(547\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(548\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(549\) 0 0
\(550\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(551\) 2.00000 2.00000
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −2.00000 2.00000i −2.00000 2.00000i
\(563\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 1.00000i 1.00000 1.00000i
\(576\) 1.00000 1.00000
\(577\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(578\) 1.00000 1.00000i 1.00000 1.00000i
\(579\) 0 0
\(580\) 2.00000i 2.00000i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(586\) 2.00000 2.00000
\(587\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 2.00000 2.00000i 2.00000 2.00000i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.00000i 2.00000i
\(605\) 1.00000i 1.00000i
\(606\) 0 0
\(607\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(608\) 1.00000 1.00000i 1.00000 1.00000i
\(609\) 0 0
\(610\) 0 0
\(611\) −2.00000 −2.00000
\(612\) 0 0
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 2.00000i 2.00000i
\(615\) 0 0
\(616\) 0 0
\(617\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 1.00000
\(626\) −2.00000 −2.00000
\(627\) 0 0
\(628\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(636\) 0 0
\(637\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(638\) −2.00000 + 2.00000i −2.00000 + 2.00000i
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.00000 2.00000
\(651\) 0 0
\(652\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(653\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.00000 + 2.00000i 2.00000 + 2.00000i
\(668\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(674\) 2.00000i 2.00000i
\(675\) 0 0
\(676\) 1.00000 1.00000
\(677\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(684\) 1.00000i 1.00000i
\(685\) 1.00000 1.00000i 1.00000 1.00000i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000i 1.00000i
\(705\) 0 0
\(706\) 2.00000 2.00000
\(707\) 0 0
\(708\) 0 0
\(709\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −1.00000 1.00000i −1.00000 1.00000i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −1.00000 −1.00000
\(721\) 0 0
\(722\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000i 2.00000i
\(726\) 0 0
\(727\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 2.00000i 2.00000i
\(735\) 0 0
\(736\) 2.00000 2.00000
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.00000 2.00000
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −1.00000 1.00000i −1.00000 1.00000i
\(753\) 0 0
\(754\) 4.00000i 4.00000i
\(755\) 2.00000i 2.00000i
\(756\) 0 0
\(757\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000i 1.00000i
\(785\) 1.00000 1.00000i 1.00000 1.00000i
\(786\) 0 0
\(787\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 2.00000i 2.00000i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0