Properties

Label 693.1.bp.a.629.3
Level $693$
Weight $1$
Character 693.629
Analytic conductor $0.346$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,1,Mod(62,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.62");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 693.bp (of order \(10\), degree \(4\), 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: \(0.345852053755\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

Embedding invariants

Embedding label 629.3
Root \(-0.891007 - 0.453990i\) of defining polynomial
Character \(\chi\) \(=\) 693.629
Dual form 693.1.bp.a.314.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.0966818 - 0.297556i) q^{2} +(0.729825 + 0.530249i) q^{4} +(-0.587785 + 0.809017i) q^{7} +(0.481456 - 0.349798i) q^{8} +O(q^{10})\) \(q+(0.0966818 - 0.297556i) q^{2} +(0.729825 + 0.530249i) q^{4} +(-0.587785 + 0.809017i) q^{7} +(0.481456 - 0.349798i) q^{8} +(0.453990 - 0.891007i) q^{11} +(0.183900 + 0.253116i) q^{14} +(0.221232 + 0.680881i) q^{16} +(-0.221232 - 0.221232i) q^{22} +1.97538i q^{23} +(0.809017 - 0.587785i) q^{25} +(-0.857960 + 0.278768i) q^{28} +(-1.44168 - 1.04744i) q^{29} +0.819101 q^{32} +(-0.951057 - 0.690983i) q^{37} -0.618034i q^{43} +(0.803789 - 0.409551i) q^{44} +(0.587785 + 0.190983i) q^{46} +(-0.309017 - 0.951057i) q^{49} +(-0.0966818 - 0.297556i) q^{50} +(-1.69480 - 0.550672i) q^{53} +0.595112i q^{56} +(-0.451057 + 0.327712i) q^{58} +(-0.142040 + 0.437153i) q^{64} +0.618034 q^{67} +(0.863541 - 0.280582i) q^{71} +(-0.297556 + 0.216187i) q^{74} +(0.453990 + 0.891007i) q^{77} +(-1.80902 - 0.587785i) q^{79} +(-0.183900 - 0.0597526i) q^{86} +(-0.0930960 - 0.587785i) q^{88} +(-1.04744 + 1.44168i) q^{92} -0.312869 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{4} + 4 q^{16} - 4 q^{22} + 4 q^{25} - 20 q^{28} + 4 q^{49} + 8 q^{58} + 4 q^{64} - 8 q^{67} - 20 q^{79} + 20 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0966818 0.297556i 0.0966818 0.297556i −0.891007 0.453990i \(-0.850000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(3\) 0 0
\(4\) 0.729825 + 0.530249i 0.729825 + 0.530249i
\(5\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(6\) 0 0
\(7\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(8\) 0.481456 0.349798i 0.481456 0.349798i
\(9\) 0 0
\(10\) 0 0
\(11\) 0.453990 0.891007i 0.453990 0.891007i
\(12\) 0 0
\(13\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(14\) 0.183900 + 0.253116i 0.183900 + 0.253116i
\(15\) 0 0
\(16\) 0.221232 + 0.680881i 0.221232 + 0.680881i
\(17\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(18\) 0 0
\(19\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.221232 0.221232i −0.221232 0.221232i
\(23\) 1.97538i 1.97538i 0.156434 + 0.987688i \(0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(24\) 0 0
\(25\) 0.809017 0.587785i 0.809017 0.587785i
\(26\) 0 0
\(27\) 0 0
\(28\) −0.857960 + 0.278768i −0.857960 + 0.278768i
\(29\) −1.44168 1.04744i −1.44168 1.04744i −0.987688 0.156434i \(-0.950000\pi\)
−0.453990 0.891007i \(-0.650000\pi\)
\(30\) 0 0
\(31\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(32\) 0.819101 0.819101
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.951057 0.690983i −0.951057 0.690983i 1.00000i \(-0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(42\) 0 0
\(43\) 0.618034i 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(44\) 0.803789 0.409551i 0.803789 0.409551i
\(45\) 0 0
\(46\) 0.587785 + 0.190983i 0.587785 + 0.190983i
\(47\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(48\) 0 0
\(49\) −0.309017 0.951057i −0.309017 0.951057i
\(50\) −0.0966818 0.297556i −0.0966818 0.297556i
\(51\) 0 0
\(52\) 0 0
\(53\) −1.69480 0.550672i −1.69480 0.550672i −0.707107 0.707107i \(-0.750000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.595112i 0.595112i
\(57\) 0 0
\(58\) −0.451057 + 0.327712i −0.451057 + 0.327712i
\(59\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(60\) 0 0
\(61\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.142040 + 0.437153i −0.142040 + 0.437153i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.863541 0.280582i 0.863541 0.280582i 0.156434 0.987688i \(-0.450000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) 0 0
\(73\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(74\) −0.297556 + 0.216187i −0.297556 + 0.216187i
\(75\) 0 0
\(76\) 0 0
\(77\) 0.453990 + 0.891007i 0.453990 + 0.891007i
\(78\) 0 0
\(79\) −1.80902 0.587785i −1.80902 0.587785i −0.809017 0.587785i \(-0.800000\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.183900 0.0597526i −0.183900 0.0597526i
\(87\) 0 0
\(88\) −0.0930960 0.587785i −0.0930960 0.587785i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.04744 + 1.44168i −1.04744 + 1.44168i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(98\) −0.312869 −0.312869
\(99\) 0 0
\(100\) 0.902113 0.902113
\(101\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −0.327712 + 0.451057i −0.327712 + 0.451057i
\(107\) −1.59811 + 1.16110i −1.59811 + 1.16110i −0.707107 + 0.707107i \(0.750000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(108\) 0 0
\(109\) 1.61803i 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.680881 0.221232i −0.680881 0.221232i
\(113\) 0.533698 + 0.734572i 0.533698 + 0.734572i 0.987688 0.156434i \(-0.0500000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.496769 1.52890i −0.496769 1.52890i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.587785 0.809017i −0.587785 0.809017i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1.11803 + 0.363271i −1.11803 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(128\) 0.779012 + 0.565985i 0.779012 + 0.565985i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.0597526 0.183900i 0.0597526 0.183900i
\(135\) 0 0
\(136\) 0 0
\(137\) 0.297556 0.0966818i 0.297556 0.0966818i −0.156434 0.987688i \(-0.550000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(138\) 0 0
\(139\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.284079i 0.284079i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −0.327712 1.00859i −0.327712 1.00859i
\(149\) 0.437016 + 1.34500i 0.437016 + 1.34500i 0.891007 + 0.453990i \(0.150000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(150\) 0 0
\(151\) 0.951057 + 1.30902i 0.951057 + 1.30902i 0.951057 + 0.309017i \(0.100000\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0.309017 0.0489435i 0.309017 0.0489435i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(158\) −0.349798 + 0.481456i −0.349798 + 0.481456i
\(159\) 0 0
\(160\) 0 0
\(161\) −1.59811 1.16110i −1.59811 1.16110i
\(162\) 0 0
\(163\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(168\) 0 0
\(169\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(170\) 0 0
\(171\) 0 0
\(172\) 0.327712 0.451057i 0.327712 0.451057i
\(173\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(174\) 0 0
\(175\) 1.00000i 1.00000i
\(176\) 0.707107 + 0.111995i 0.707107 + 0.111995i
\(177\) 0 0
\(178\) 0 0
\(179\) −0.183900 0.253116i −0.183900 0.253116i 0.707107 0.707107i \(-0.250000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(180\) 0 0
\(181\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.690983 + 0.951057i 0.690983 + 0.951057i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.04744 1.44168i 1.04744 1.44168i 0.156434 0.987688i \(-0.450000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(192\) 0 0
\(193\) 0.587785 0.190983i 0.587785 0.190983i 1.00000i \(-0.5\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.278768 0.857960i 0.278768 0.857960i
\(197\) 1.97538 1.97538 0.987688 0.156434i \(-0.0500000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0.183900 0.565985i 0.183900 0.565985i
\(201\) 0 0
\(202\) 0 0
\(203\) 1.69480 0.550672i 1.69480 0.550672i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.53884 + 0.500000i 1.53884 + 0.500000i 0.951057 0.309017i \(-0.100000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(212\) −0.944910 1.30056i −0.944910 1.30056i
\(213\) 0 0
\(214\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.481456 0.156434i −0.481456 0.156434i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(224\) −0.481456 + 0.662667i −0.481456 + 0.662667i
\(225\) 0 0
\(226\) 0.270175 0.0877853i 0.270175 0.0877853i
\(227\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(228\) 0 0
\(229\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.06050 −1.06050
\(233\) 0.437016 1.34500i 0.437016 1.34500i −0.453990 0.891007i \(-0.650000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.734572 + 0.533698i −0.734572 + 0.533698i −0.891007 0.453990i \(-0.850000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −0.297556 + 0.0966818i −0.297556 + 0.0966818i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(252\) 0 0
\(253\) 1.76007 + 0.896802i 1.76007 + 0.896802i
\(254\) 0.367799i 0.367799i
\(255\) 0 0
\(256\) −0.128136 + 0.0930960i −0.128136 + 0.0930960i
\(257\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(258\) 0 0
\(259\) 1.11803 0.363271i 1.11803 0.363271i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.78201 1.78201 0.891007 0.453990i \(-0.150000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0.451057 + 0.327712i 0.451057 + 0.327712i
\(269\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(270\) 0 0
\(271\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.0978870i 0.0978870i
\(275\) −0.156434 0.987688i −0.156434 0.987688i
\(276\) 0 0
\(277\) −1.11803 0.363271i −1.11803 0.363271i −0.309017 0.951057i \(-0.600000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.280582 0.863541i −0.280582 0.863541i −0.987688 0.156434i \(-0.950000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(284\) 0.779012 + 0.253116i 0.779012 + 0.253116i
\(285\) 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 0
\(293\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.699596 −0.699596
\(297\) 0 0
\(298\) 0.442463 0.442463
\(299\) 0 0
\(300\) 0 0
\(301\) 0.500000 + 0.363271i 0.500000 + 0.363271i
\(302\) 0.481456 0.156434i 0.481456 0.156434i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −0.141122 + 0.891007i −0.141122 + 0.891007i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(312\) 0 0
\(313\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.00859 1.38821i −1.00859 1.38821i
\(317\) −0.863541 0.280582i −0.863541 0.280582i −0.156434 0.987688i \(-0.550000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) −1.58779 + 0.809017i −1.58779 + 0.809017i
\(320\) 0 0
\(321\) 0 0
\(322\) −0.500000 + 0.363271i −0.500000 + 0.363271i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0.409551 + 0.297556i 0.409551 + 0.297556i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.17557 1.17557 0.587785 0.809017i \(-0.300000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.951057 + 1.30902i −0.951057 + 1.30902i 1.00000i \(0.5\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(338\) 0.253116 0.183900i 0.253116 0.183900i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.951057 + 0.309017i 0.951057 + 0.309017i
\(344\) −0.216187 0.297556i −0.216187 0.297556i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.280582 + 0.863541i 0.280582 + 0.863541i 0.987688 + 0.156434i \(0.0500000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(350\) 0.297556 + 0.0966818i 0.297556 + 0.0966818i
\(351\) 0 0
\(352\) 0.371864 0.729825i 0.371864 0.729825i
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −0.0930960 + 0.0302487i −0.0930960 + 0.0302487i
\(359\) 1.59811 + 1.16110i 1.59811 + 1.16110i 0.891007 + 0.453990i \(0.150000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(360\) 0 0
\(361\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(368\) −1.34500 + 0.437016i −1.34500 + 0.437016i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.44168 1.04744i 1.44168 1.04744i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.363271 1.11803i −0.363271 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.327712 0.451057i −0.327712 0.451057i
\(383\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.193364i 0.193364i
\(387\) 0 0
\(388\) 0 0
\(389\) −0.183900 + 0.253116i −0.183900 + 0.253116i −0.891007 0.453990i \(-0.850000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.481456 0.349798i −0.481456 0.349798i
\(393\) 0 0
\(394\) 0.190983 0.587785i 0.190983 0.587785i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.579192 + 0.420808i 0.579192 + 0.420808i
\(401\) −1.87869 + 0.610425i −1.87869 + 0.610425i −0.891007 + 0.453990i \(0.850000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0.557537i 0.557537i
\(407\) −1.04744 + 0.533698i −1.04744 + 0.533698i
\(408\) 0 0
\(409\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 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
\(421\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(422\) 0.297556 0.409551i 0.297556 0.409551i
\(423\) 0 0
\(424\) −1.00859 + 0.327712i −1.00859 + 0.327712i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.78201 −1.78201
\(429\) 0 0
\(430\) 0 0
\(431\) −0.610425 + 1.87869i −0.610425 + 1.87869i −0.156434 + 0.987688i \(0.550000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(432\) 0 0
\(433\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.857960 1.18088i 0.857960 1.18088i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.533698 0.734572i −0.533698 0.734572i 0.453990 0.891007i \(-0.350000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.270175 0.371864i −0.270175 0.371864i
\(449\) 1.87869 + 0.610425i 1.87869 + 0.610425i 0.987688 + 0.156434i \(0.0500000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.819101i 0.819101i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.53884 0.500000i 1.53884 0.500000i 0.587785 0.809017i \(-0.300000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(464\) 0.394238 1.21334i 0.394238 1.21334i
\(465\) 0 0
\(466\) −0.357960 0.260074i −0.357960 0.260074i
\(467\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(468\) 0 0
\(469\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.550672 0.280582i −0.550672 0.280582i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0.0877853 + 0.270175i 0.0877853 + 0.270175i
\(479\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.902113i 0.902113i
\(485\) 0 0
\(486\) 0 0
\(487\) −1.53884 + 1.11803i −1.53884 + 1.11803i −0.587785 + 0.809017i \(0.700000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.14412 0.831254i −1.14412 0.831254i −0.156434 0.987688i \(-0.550000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.280582 + 0.863541i −0.280582 + 0.863541i
\(498\) 0 0
\(499\) −1.53884 1.11803i −1.53884 1.11803i −0.951057 0.309017i \(-0.900000\pi\)
−0.587785 0.809017i \(-0.700000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0.437016 0.437016i 0.437016 0.437016i
\(507\) 0 0
\(508\) −1.00859 0.327712i −1.00859 0.327712i
\(509\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.312869 + 0.962912i 0.312869 + 0.962912i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0.367799i 0.367799i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(522\) 0 0
\(523\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0.172288 0.530249i 0.172288 0.530249i
\(527\) 0 0
\(528\) 0 0
\(529\) −2.90211 −2.90211
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.297556 0.216187i 0.297556 0.216187i
\(537\) 0 0
\(538\) 0 0
\(539\) −0.987688 0.156434i −0.987688 0.156434i
\(540\) 0 0
\(541\) 1.11803 + 0.363271i 1.11803 + 0.363271i 0.809017 0.587785i \(-0.200000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(548\) 0.268429 + 0.0872179i 0.268429 + 0.0872179i
\(549\) 0 0
\(550\) −0.309017 0.0489435i −0.309017 0.0489435i
\(551\) 0 0
\(552\) 0 0
\(553\) 1.53884 1.11803i 1.53884 1.11803i
\(554\) −0.216187 + 0.297556i −0.216187 + 0.297556i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.734572 + 0.533698i 0.734572 + 0.533698i 0.891007 0.453990i \(-0.150000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.284079 −0.284079
\(563\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0.317610 0.437153i 0.317610 0.437153i
\(569\) −1.14412 + 0.831254i −1.14412 + 0.831254i −0.987688 0.156434i \(-0.950000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(570\) 0 0
\(571\) 1.17557i 1.17557i 0.809017 + 0.587785i \(0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.16110 + 1.59811i 1.16110 + 1.59811i
\(576\) 0 0
\(577\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(578\) 0.0966818 + 0.297556i 0.0966818 + 0.297556i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.26007 + 1.26007i −1.26007 + 1.26007i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.260074 0.800424i 0.260074 0.800424i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.394238 + 1.21334i −0.394238 + 1.21334i
\(597\) 0 0
\(598\) 0 0
\(599\) −0.297556 + 0.0966818i −0.297556 + 0.0966818i −0.453990 0.891007i \(-0.650000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(600\) 0 0
\(601\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(602\) 0.156434 0.113656i 0.156434 0.113656i
\(603\) 0 0
\(604\) 1.45965i 1.45965i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.11803 1.53884i −1.11803 1.53884i −0.809017 0.587785i \(-0.800000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.530249 + 0.270175i 0.530249 + 0.270175i
\(617\) 1.78201i 1.78201i −0.453990 0.891007i \(-0.650000\pi\)
0.453990 0.891007i \(-0.350000\pi\)
\(618\) 0 0
\(619\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(620\) 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
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.30902 0.951057i −1.30902 0.951057i −0.309017 0.951057i \(-0.600000\pi\)
−1.00000 \(\pi\)
\(632\) −1.07657 + 0.349798i −1.07657 + 0.349798i
\(633\) 0 0
\(634\) −0.166977 + 0.229825i −0.166977 + 0.229825i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0.0872179 + 0.550672i 0.0872179 + 0.550672i
\(639\) 0 0
\(640\) 0 0
\(641\) 1.04744 + 1.44168i 1.04744 + 1.44168i 0.891007 + 0.453990i \(0.150000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(642\) 0 0
\(643\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(644\) −0.550672 1.69480i −0.550672 1.69480i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.18088 + 0.857960i −1.18088 + 0.857960i
\(653\) 1.16110 1.59811i 1.16110 1.59811i 0.453990 0.891007i \(-0.350000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0.113656 0.349798i 0.113656 0.349798i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.06909 2.84786i 2.06909 2.84786i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.587785 0.190983i −0.587785 0.190983i 1.00000i \(-0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(674\) 0.297556 + 0.409551i 0.297556 + 0.409551i
\(675\) 0 0
\(676\) 0.278768 + 0.857960i 0.278768 + 0.857960i
\(677\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.312869i 0.312869i −0.987688 0.156434i \(-0.950000\pi\)
0.987688 0.156434i \(-0.0500000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.183900 0.253116i 0.183900 0.253116i
\(687\) 0 0
\(688\) 0.420808 0.136729i 0.420808 0.136729i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0.284079 0.284079
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.530249 + 0.729825i −0.530249 + 0.729825i
\(701\) 1.59811 1.16110i 1.59811 1.16110i 0.707107 0.707107i \(-0.250000\pi\)
0.891007 0.453990i \(-0.150000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0.325021 + 0.325021i 0.325021 + 0.325021i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.587785 + 1.80902i 0.587785 + 1.80902i 0.587785 + 0.809017i \(0.300000\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.282243i 0.282243i
\(717\) 0 0
\(718\) 0.500000 0.363271i 0.500000 0.363271i
\(719\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.253116 + 0.183900i 0.253116 + 0.183900i
\(723\) 0 0
\(724\) 0 0
\(725\) −1.78201 −1.78201
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.61803i 1.61803i
\(737\) 0.280582 0.550672i 0.280582 0.550672i
\(738\) 0 0
\(739\) 1.80902 + 0.587785i 1.80902 + 0.587785i 1.00000 \(0\)
0.809017 + 0.587785i \(0.200000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.172288 0.530249i −0.172288 0.530249i
\(743\) −0.550672 1.69480i −0.550672 1.69480i −0.707107 0.707107i \(-0.750000\pi\)
0.156434 0.987688i \(-0.450000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.97538i 1.97538i
\(750\) 0 0
\(751\) 0.951057 0.690983i 0.951057 0.690983i 1.00000i \(-0.5\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.587785 + 1.80902i −0.587785 + 1.80902i 1.00000i \(0.5\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(758\) −0.367799 −0.367799
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(762\) 0 0
\(763\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(764\) 1.52890 0.496769i 1.52890 0.496769i
\(765\) 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
\(772\) 0.530249 + 0.172288i 0.530249 + 0.172288i
\(773\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0.0575365 + 0.0791922i 0.0575365 + 0.0791922i
\(779\) 0 0
\(780\) 0 0
\(781\) 0.142040 0.896802i 0.142040 0.896802i
\(782\) 0 0
\(783\) 0 0
\(784\) 0.579192 0.420808i 0.579192 0.420808i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(788\) 1.44168 + 1.04744i 1.44168 + 1.04744i
\(789\) 0 0
\(790\) 0 0
\(791\) −0.907981 −0.907981
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(798\) 0 0
\(799\) 0 0