Properties

Label 3267.1.l.a.2998.4
Level $3267$
Weight $1$
Character 3267.2998
Analytic conductor $1.630$
Analytic rank $0$
Dimension $16$
Projective image $D_{12}$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3267,1,Mod(838,3267)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3267, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3267.838");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3267.l (of order \(10\), 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: \(1.63044539627\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 2998.4
Root \(1.83730 - 0.596975i\) of defining polynomial
Character \(\chi\) \(=\) 3267.2998
Dual form 3267.1.l.a.838.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.809017 + 0.587785i) q^{4} +(1.13551 + 1.56290i) q^{7} +O(q^{10})\) \(q+(-0.809017 + 0.587785i) q^{4} +(1.13551 + 1.56290i) q^{7} +(0.492303 - 0.159959i) q^{13} +(0.309017 - 0.951057i) q^{16} +(0.831254 - 1.14412i) q^{19} +(0.809017 + 0.587785i) q^{25} +(-1.83730 - 0.596975i) q^{28} +(0.535233 + 1.64728i) q^{31} -1.41421i q^{43} +(-0.844250 + 2.59833i) q^{49} +(-0.304260 + 0.418778i) q^{52} +(-1.34500 - 0.437016i) q^{61} +(0.309017 + 0.951057i) q^{64} -1.73205 q^{67} +(0.304260 + 0.418778i) q^{73} +1.41421i q^{76} +(-0.492303 + 0.159959i) q^{79} +(0.809017 + 0.587785i) q^{91} +(0.309017 + 0.951057i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} - 4 q^{16} + 4 q^{25} + 4 q^{49} - 4 q^{64} + 4 q^{91} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(3026\)
\(\chi(n)\) \(e\left(\frac{9}{10}\right)\) \(1\)

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 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(3\) 0 0
\(4\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(5\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(6\) 0 0
\(7\) 1.13551 + 1.56290i 1.13551 + 1.56290i 0.777146 + 0.629320i \(0.216667\pi\)
0.358368 + 0.933580i \(0.383333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0.492303 0.159959i 0.492303 0.159959i −0.0523360 0.998630i \(-0.516667\pi\)
0.544639 + 0.838671i \(0.316667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.309017 0.951057i 0.309017 0.951057i
\(17\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(18\) 0 0
\(19\) 0.831254 1.14412i 0.831254 1.14412i −0.156434 0.987688i \(-0.550000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(26\) 0 0
\(27\) 0 0
\(28\) −1.83730 0.596975i −1.83730 0.596975i
\(29\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(30\) 0 0
\(31\) 0.535233 + 1.64728i 0.535233 + 1.64728i 0.743145 + 0.669131i \(0.233333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(42\) 0 0
\(43\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(48\) 0 0
\(49\) −0.844250 + 2.59833i −0.844250 + 2.59833i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.304260 + 0.418778i −0.304260 + 0.418778i
\(53\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(60\) 0 0
\(61\) −1.34500 0.437016i −1.34500 0.437016i −0.453990 0.891007i \(-0.650000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(72\) 0 0
\(73\) 0.304260 + 0.418778i 0.304260 + 0.418778i 0.933580 0.358368i \(-0.116667\pi\)
−0.629320 + 0.777146i \(0.716667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.41421i 1.41421i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.492303 + 0.159959i −0.492303 + 0.159959i −0.544639 0.838671i \(-0.683333\pi\)
0.0523360 + 0.998630i \(0.483333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.309017 + 0.951057i 0.309017 + 0.951057i 0.978148 + 0.207912i \(0.0666667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −1.00000
\(101\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(102\) 0 0
\(103\) −0.809017 + 0.587785i −0.809017 + 0.587785i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(108\) 0 0
\(109\) 0.517638i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.83730 0.596975i 1.83730 0.596975i
\(113\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −1.40126 1.01807i −1.40126 1.01807i
\(125\) 0 0
\(126\) 0 0
\(127\) 1.83730 + 0.596975i 1.83730 + 0.596975i 0.998630 + 0.0523360i \(0.0166667\pi\)
0.838671 + 0.544639i \(0.183333\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 2.73205 2.73205
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(138\) 0 0
\(139\) −1.13551 1.56290i −1.13551 1.56290i −0.777146 0.629320i \(-0.783333\pi\)
−0.358368 0.933580i \(-0.616667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(150\) 0 0
\(151\) 1.13551 1.56290i 1.13551 1.56290i 0.358368 0.933580i \(-0.383333\pi\)
0.777146 0.629320i \(-0.216667\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.40126 + 1.01807i 1.40126 + 1.01807i 0.994522 + 0.104528i \(0.0333333\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(168\) 0 0
\(169\) −0.592242 + 0.430289i −0.592242 + 0.430289i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.831254 + 1.14412i 0.831254 + 1.14412i
\(173\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(174\) 0 0
\(175\) 1.93185i 1.93185i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(180\) 0 0
\(181\) 0.309017 0.951057i 0.309017 0.951057i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(192\) 0 0
\(193\) 1.34500 + 0.437016i 1.34500 + 0.437016i 0.891007 0.453990i \(-0.150000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.844250 2.59833i −0.844250 2.59833i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.517638i 0.517638i
\(209\) 0 0
\(210\) 0 0
\(211\) −1.34500 + 0.437016i −1.34500 + 0.437016i −0.891007 0.453990i \(-0.850000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.96677 + 2.70702i −1.96677 + 2.70702i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −0.809017 0.587785i −0.809017 0.587785i 0.104528 0.994522i \(-0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(228\) 0 0
\(229\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(240\) 0 0
\(241\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 1.34500 0.437016i 1.34500 0.437016i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.226216 0.696222i 0.226216 0.696222i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.809017 0.587785i −0.809017 0.587785i
\(257\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.40126 1.01807i 1.40126 1.01807i
\(269\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(270\) 0 0
\(271\) −0.304260 0.418778i −0.304260 0.418778i 0.629320 0.777146i \(-0.283333\pi\)
−0.933580 + 0.358368i \(0.883333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.83730 + 0.596975i −1.83730 + 0.596975i −0.838671 + 0.544639i \(0.816667\pi\)
−0.998630 + 0.0523360i \(0.983333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(282\) 0 0
\(283\) 0.304260 0.418778i 0.304260 0.418778i −0.629320 0.777146i \(-0.716667\pi\)
0.933580 + 0.358368i \(0.116667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.809017 0.587785i −0.809017 0.587785i
\(290\) 0 0
\(291\) 0 0
\(292\) −0.492303 0.159959i −0.492303 0.159959i
\(293\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 2.21028 1.60586i 2.21028 1.60586i
\(302\) 0 0
\(303\) 0 0
\(304\) −0.831254 1.14412i −0.831254 1.14412i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(312\) 0 0
\(313\) 0.535233 1.64728i 0.535233 1.64728i −0.207912 0.978148i \(-0.566667\pi\)
0.743145 0.669131i \(-0.233333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.304260 0.418778i 0.304260 0.418778i
\(317\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0.492303 + 0.159959i 0.492303 + 0.159959i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.831254 1.14412i −0.831254 1.14412i −0.987688 0.156434i \(-0.950000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −3.18230 + 1.03399i −3.18230 + 1.03399i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(348\) 0 0
\(349\) −0.831254 + 1.14412i −0.831254 + 1.14412i 0.156434 + 0.987688i \(0.450000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(360\) 0 0
\(361\) −0.309017 0.951057i −0.309017 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) −1.00000 −1.00000
\(365\) 0 0
\(366\) 0 0
\(367\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.517638i 0.517638i −0.965926 0.258819i \(-0.916667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −0.809017 0.587785i −0.809017 0.587785i
\(389\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.809017 0.587785i 0.809017 0.587785i
\(401\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(402\) 0 0
\(403\) 0.526994 + 0.725345i 0.526994 + 0.725345i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.34500 0.437016i 1.34500 0.437016i 0.453990 0.891007i \(-0.350000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.309017 0.951057i 0.309017 0.951057i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 1.40126 + 1.01807i 1.40126 + 1.01807i 0.994522 + 0.104528i \(0.0333333\pi\)
0.406737 + 0.913545i \(0.366667\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.844250 2.59833i −0.844250 2.59833i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(432\) 0 0
\(433\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.304260 0.418778i −0.304260 0.418778i
\(437\) 0 0
\(438\) 0 0
\(439\) 1.93185i 1.93185i 0.258819 + 0.965926i \(0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.13551 + 1.56290i −1.13551 + 1.56290i
\(449\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.34500 + 0.437016i 1.34500 + 0.437016i 0.891007 0.453990i \(-0.150000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(468\) 0 0
\(469\) −1.96677 2.70702i −1.96677 2.70702i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.34500 0.437016i 1.34500 0.437016i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.40126 1.01807i −1.40126 1.01807i −0.994522 0.104528i \(-0.966667\pi\)
−0.406737 0.913545i \(-0.633333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.73205 1.73205
\(497\) 0 0
\(498\) 0 0
\(499\) −1.40126 + 1.01807i −1.40126 + 1.01807i −0.406737 + 0.913545i \(0.633333\pi\)
−0.994522 + 0.104528i \(0.966667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −1.83730 + 0.596975i −1.83730 + 0.596975i
\(509\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(510\) 0 0
\(511\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(522\) 0 0
\(523\) 1.34500 + 0.437016i 1.34500 + 0.437016i 0.891007 0.453990i \(-0.150000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −2.21028 + 1.60586i −2.21028 + 1.60586i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.83730 0.596975i 1.83730 0.596975i 0.838671 0.544639i \(-0.183333\pi\)
0.998630 0.0523360i \(-0.0166667\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.304260 0.418778i 0.304260 0.418778i −0.629320 0.777146i \(-0.716667\pi\)
0.933580 + 0.358368i \(0.116667\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.809017 0.587785i −0.809017 0.587785i
\(554\) 0 0
\(555\) 0 0
\(556\) 1.83730 + 0.596975i 1.83730 + 0.596975i
\(557\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(558\) 0 0
\(559\) −0.226216 0.696222i −0.226216 0.696222i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(570\) 0 0
\(571\) 0.517638i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(588\) 0 0
\(589\) 2.32960 + 0.756934i 2.32960 + 0.756934i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(600\) 0 0
\(601\) 0.304260 + 0.418778i 0.304260 + 0.418778i 0.933580 0.358368i \(-0.116667\pi\)
−0.629320 + 0.777146i \(0.716667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.93185i 1.93185i
\(605\) 0 0
\(606\) 0 0
\(607\) −0.492303 + 0.159959i −0.492303 + 0.159959i −0.544639 0.838671i \(-0.683333\pi\)
0.0523360 + 0.998630i \(0.483333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.13551 1.56290i 1.13551 1.56290i 0.358368 0.933580i \(-0.383333\pi\)
0.777146 0.629320i \(-0.216667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.73205 −1.73205
\(629\) 0 0
\(630\) 0 0
\(631\) 1.40126 1.01807i 1.40126 1.01807i 0.406737 0.913545i \(-0.366667\pi\)
0.994522 0.104528i \(-0.0333333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.41421i 1.41421i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(642\) 0 0
\(643\) 0.309017 0.951057i 0.309017 0.951057i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(653\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) 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
\(673\) 1.83730 0.596975i 1.83730 0.596975i 0.838671 0.544639i \(-0.183333\pi\)
0.998630 0.0523360i \(-0.0166667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.226216 0.696222i 0.226216 0.696222i
\(677\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(678\) 0 0
\(679\) −1.13551 + 1.56290i −1.13551 + 1.56290i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −1.34500 0.437016i −1.34500 0.437016i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.13551 1.56290i −1.13551 1.56290i
\(701\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.535233 + 1.64728i −0.535233 + 1.64728i 0.207912 + 0.978148i \(0.433333\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(720\) 0 0
\(721\) −1.83730 0.596975i −1.83730 0.596975i
\(722\) 0 0
\(723\) 0 0
\(724\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0.831254 + 1.14412i 0.831254 + 1.14412i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.492303 + 0.159959i −0.492303 + 0.159959i −0.544639 0.838671i \(-0.683333\pi\)
0.0523360 + 0.998630i \(0.483333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.40126 1.01807i −1.40126 1.01807i −0.994522 0.104528i \(-0.966667\pi\)
−0.406737 0.913545i \(-0.633333\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.535233 + 1.64728i 0.535233 + 1.64728i 0.743145 + 0.669131i \(0.233333\pi\)
−0.207912 + 0.978148i \(0.566667\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(762\) 0 0
\(763\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.93185i 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.34500 + 0.437016i −1.34500 + 0.437016i
\(773\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(774\) 0 0
\(775\) −0.535233 + 1.64728i −0.535233 + 1.64728i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.21028 + 1.60586i 2.21028 + 1.60586i
\(785\) 0 0
\(786\) 0 0
\(787\) −1.83730 0.596975i −1.83730 0.596975i −0.998630 0.0523360i \(-0.983333\pi\)
−0.838671 0.544639i \(-0.816667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.732051 −0.732051
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\)