Properties

Label 804.1.be.a.629.1
Level $804$
Weight $1$
Character 804.629
Analytic conductor $0.401$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [804,1,Mod(17,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 33, 64]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.17");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 804.be (of order \(66\), degree \(20\), 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: \(0.401248270157\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{33}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

Embedding invariants

Embedding label 629.1
Root \(-0.327068 - 0.945001i\) of defining polynomial
Character \(\chi\) \(=\) 804.629
Dual form 804.1.be.a.317.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.841254 - 0.540641i) q^{3} +(1.30379 - 0.124497i) q^{7} +(0.415415 - 0.909632i) q^{9} +O(q^{10})\) \(q+(0.841254 - 0.540641i) q^{3} +(1.30379 - 0.124497i) q^{7} +(0.415415 - 0.909632i) q^{9} +(-1.44091 + 1.37391i) q^{13} +(-1.67489 - 0.159932i) q^{19} +(1.02951 - 0.809616i) q^{21} +(-0.959493 - 0.281733i) q^{25} +(-0.142315 - 0.989821i) q^{27} +(1.34378 + 1.28129i) q^{31} +(0.959493 - 1.66189i) q^{37} +(-0.469383 + 1.93482i) q^{39} +(-0.308779 + 0.356349i) q^{43} +(0.702443 - 0.135385i) q^{49} +(-1.49547 + 0.770969i) q^{57} +(-1.65033 - 0.660694i) q^{61} +(0.428368 - 1.23769i) q^{63} +(0.235759 + 0.971812i) q^{67} +(-1.45949 - 0.584293i) q^{73} +(-0.959493 + 0.281733i) q^{75} +(0.341254 + 1.40667i) q^{79} +(-0.654861 - 0.755750i) q^{81} +(-1.70760 + 1.97068i) q^{91} +(1.82318 + 0.351390i) q^{93} +(-0.0475819 + 0.0824143i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} + 2 q^{7} - 2 q^{9} - q^{13} + 2 q^{19} + 2 q^{21} - 2 q^{25} - 2 q^{27} - q^{31} + 2 q^{37} - q^{39} + 2 q^{43} + 3 q^{49} - 9 q^{57} - q^{61} - 9 q^{63} + q^{67} - 12 q^{73} - 2 q^{75} - 12 q^{79} - 2 q^{81} + 4 q^{91} - q^{93} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{10}{33}\right)\) \(1\)

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
\(3\) 0.841254 0.540641i 0.841254 0.540641i
\(4\) 0 0
\(5\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(6\) 0 0
\(7\) 1.30379 0.124497i 1.30379 0.124497i 0.580057 0.814576i \(-0.303030\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(8\) 0 0
\(9\) 0.415415 0.909632i 0.415415 0.909632i
\(10\) 0 0
\(11\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(12\) 0 0
\(13\) −1.44091 + 1.37391i −1.44091 + 1.37391i −0.654861 + 0.755750i \(0.727273\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(18\) 0 0
\(19\) −1.67489 0.159932i −1.67489 0.159932i −0.786053 0.618159i \(-0.787879\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(20\) 0 0
\(21\) 1.02951 0.809616i 1.02951 0.809616i
\(22\) 0 0
\(23\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(24\) 0 0
\(25\) −0.959493 0.281733i −0.959493 0.281733i
\(26\) 0 0
\(27\) −0.142315 0.989821i −0.142315 0.989821i
\(28\) 0 0
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 1.34378 + 1.28129i 1.34378 + 1.28129i 0.928368 + 0.371662i \(0.121212\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.959493 1.66189i 0.959493 1.66189i 0.235759 0.971812i \(-0.424242\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(38\) 0 0
\(39\) −0.469383 + 1.93482i −0.469383 + 1.93482i
\(40\) 0 0
\(41\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(42\) 0 0
\(43\) −0.308779 + 0.356349i −0.308779 + 0.356349i −0.888835 0.458227i \(-0.848485\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(48\) 0 0
\(49\) 0.702443 0.135385i 0.702443 0.135385i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.49547 + 0.770969i −1.49547 + 0.770969i
\(58\) 0 0
\(59\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(60\) 0 0
\(61\) −1.65033 0.660694i −1.65033 0.660694i −0.654861 0.755750i \(-0.727273\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(62\) 0 0
\(63\) 0.428368 1.23769i 0.428368 1.23769i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.235759 + 0.971812i 0.235759 + 0.971812i
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(72\) 0 0
\(73\) −1.45949 0.584293i −1.45949 0.584293i −0.500000 0.866025i \(-0.666667\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(74\) 0 0
\(75\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.341254 + 1.40667i 0.341254 + 1.40667i 0.841254 + 0.540641i \(0.181818\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) −0.654861 0.755750i −0.654861 0.755750i
\(82\) 0 0
\(83\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(90\) 0 0
\(91\) −1.70760 + 1.97068i −1.70760 + 1.97068i
\(92\) 0 0
\(93\) 1.82318 + 0.351390i 1.82318 + 0.351390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.0475819 + 0.0824143i −0.0475819 + 0.0824143i −0.888835 0.458227i \(-0.848485\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(102\) 0 0
\(103\) 0.601300 + 0.573338i 0.601300 + 0.573338i 0.928368 0.371662i \(-0.121212\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(108\) 0 0
\(109\) 1.50842 + 0.442913i 1.50842 + 0.442913i 0.928368 0.371662i \(-0.121212\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(110\) 0 0
\(111\) −0.0913090 1.91681i −0.0913090 1.91681i
\(112\) 0 0
\(113\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.651174 + 1.88144i 0.651174 + 1.88144i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.723734 0.690079i 0.723734 0.690079i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −0.469383 + 0.0448206i −0.469383 + 0.0448206i −0.327068 0.945001i \(-0.606061\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(128\) 0 0
\(129\) −0.0671040 + 0.466718i −0.0671040 + 0.466718i
\(130\) 0 0
\(131\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(132\) 0 0
\(133\) −2.20362 −2.20362
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(138\) 0 0
\(139\) −0.0135432 + 0.0941952i −0.0135432 + 0.0941952i −0.995472 0.0950560i \(-0.969697\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.517738 0.493662i 0.517738 0.493662i
\(148\) 0 0
\(149\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(150\) 0 0
\(151\) −0.607279 1.75462i −0.607279 1.75462i −0.654861 0.755750i \(-0.727273\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.0395325 + 0.829889i 0.0395325 + 0.829889i 0.928368 + 0.371662i \(0.121212\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.327068 + 0.566498i 0.327068 + 0.566498i 0.981929 0.189251i \(-0.0606061\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(168\) 0 0
\(169\) 0.141026 2.96050i 0.141026 2.96050i
\(170\) 0 0
\(171\) −0.841254 + 1.45709i −0.841254 + 1.45709i
\(172\) 0 0
\(173\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(174\) 0 0
\(175\) −1.28605 0.247866i −1.28605 0.247866i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(180\) 0 0
\(181\) −0.0845850 0.0436066i −0.0845850 0.0436066i 0.415415 0.909632i \(-0.363636\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) −1.74555 + 0.336426i −1.74555 + 0.336426i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.308779 1.27280i −0.308779 1.27280i
\(190\) 0 0
\(191\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(192\) 0 0
\(193\) −1.11312 + 0.326842i −1.11312 + 0.326842i −0.786053 0.618159i \(-0.787879\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(198\) 0 0
\(199\) 1.13915 + 1.59971i 1.13915 + 1.59971i 0.723734 + 0.690079i \(0.242424\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(200\) 0 0
\(201\) 0.723734 + 0.690079i 0.723734 + 0.690079i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.39734 0.720381i 1.39734 0.720381i 0.415415 0.909632i \(-0.363636\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.91153 + 1.50324i 1.91153 + 1.50324i
\(218\) 0 0
\(219\) −1.54370 + 0.297523i −1.54370 + 0.297523i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.0800569 + 0.0514495i 0.0800569 + 0.0514495i 0.580057 0.814576i \(-0.303030\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 0 0
\(225\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(226\) 0 0
\(227\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(228\) 0 0
\(229\) 0.341254 1.40667i 0.341254 1.40667i −0.500000 0.866025i \(-0.666667\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.04758 + 0.998867i 1.04758 + 0.998867i
\(238\) 0 0
\(239\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(240\) 0 0
\(241\) −0.205996 1.43273i −0.205996 1.43273i −0.786053 0.618159i \(-0.787879\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(242\) 0 0
\(243\) −0.959493 0.281733i −0.959493 0.281733i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.63310 2.07069i 2.63310 2.07069i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(258\) 0 0
\(259\) 1.04408 2.28621i 1.04408 2.28621i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0.975950 0.627205i 0.975950 0.627205i 0.0475819 0.998867i \(-0.484848\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(272\) 0 0
\(273\) −0.371098 + 2.58104i −0.371098 + 2.58104i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.271738 + 0.595023i −0.271738 + 0.595023i −0.995472 0.0950560i \(-0.969697\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(278\) 0 0
\(279\) 1.72373 0.690079i 1.72373 0.690079i
\(280\) 0 0
\(281\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(282\) 0 0
\(283\) −0.118239 0.258908i −0.118239 0.258908i 0.841254 0.540641i \(-0.181818\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.786053 + 0.618159i −0.786053 + 0.618159i
\(290\) 0 0
\(291\) 0.00452808 + 0.0950560i 0.00452808 + 0.0950560i
\(292\) 0 0
\(293\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −0.358218 + 0.503047i −0.358218 + 0.503047i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0.462997 1.90850i 0.462997 1.90850i 0.0475819 0.998867i \(-0.484848\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(308\) 0 0
\(309\) 0.815816 + 0.157236i 0.815816 + 0.157236i
\(310\) 0 0
\(311\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(312\) 0 0
\(313\) 0.698939 + 0.449181i 0.698939 + 0.449181i 0.841254 0.540641i \(-0.181818\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.76962 0.912303i 1.76962 0.912303i
\(326\) 0 0
\(327\) 1.50842 0.442913i 1.50842 0.442913i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.581419 1.67990i 0.581419 1.67990i −0.142315 0.989821i \(-0.545455\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(332\) 0 0
\(333\) −1.11312 1.56316i −1.11312 1.56316i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.273507 + 0.384087i 0.273507 + 0.384087i 0.928368 0.371662i \(-0.121212\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.357685 + 0.105026i −0.357685 + 0.105026i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.235759 0.971812i \(-0.575758\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(348\) 0 0
\(349\) 1.30379 + 1.50465i 1.30379 + 1.50465i 0.723734 + 0.690079i \(0.242424\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(350\) 0 0
\(351\) 1.56499 + 1.23072i 1.56499 + 1.23072i
\(352\) 0 0
\(353\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(360\) 0 0
\(361\) 1.79774 + 0.346487i 1.79774 + 0.346487i
\(362\) 0 0
\(363\) 0.235759 0.971812i 0.235759 0.971812i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.0135432 + 0.284307i −0.0135432 + 0.284307i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.580057 1.00469i −0.580057 1.00469i −0.995472 0.0950560i \(-0.969697\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.0552004 + 1.15880i 0.0552004 + 1.15880i 0.841254 + 0.540641i \(0.181818\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(380\) 0 0
\(381\) −0.370638 + 0.291473i −0.370638 + 0.291473i
\(382\) 0 0
\(383\) 0 0 −0.995472 0.0950560i \(-0.969697\pi\)
0.995472 + 0.0950560i \(0.0303030\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.195876 + 0.428908i 0.195876 + 0.428908i
\(388\) 0 0
\(389\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.279486 + 1.94387i −0.279486 + 1.94387i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(398\) 0 0
\(399\) −1.85380 + 1.19136i −1.85380 + 1.19136i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −3.69666 −3.69666
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.283341 0.0270558i 0.283341 0.0270558i 0.0475819 0.998867i \(-0.484848\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.0395325 + 0.0865641i 0.0395325 + 0.0865641i
\(418\) 0 0
\(419\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(420\) 0 0
\(421\) −0.469383 0.0448206i −0.469383 0.0448206i −0.142315 0.989821i \(-0.545455\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.23394 0.655945i −2.23394 0.655945i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 0 0
\(433\) −1.38884 1.32425i −1.38884 1.32425i −0.888835 0.458227i \(-0.848485\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0.327068 0.566498i 0.327068 0.566498i −0.654861 0.755750i \(-0.727273\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(440\) 0 0
\(441\) 0.168655 0.695205i 0.168655 0.695205i
\(442\) 0 0
\(443\) 0 0 −0.981929 0.189251i \(-0.939394\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 −0.888835 0.458227i \(-0.848485\pi\)
0.888835 + 0.458227i \(0.151515\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −1.45949 1.14776i −1.45949 1.14776i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.469383 1.93482i −0.469383 1.93482i −0.327068 0.945001i \(-0.606061\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(462\) 0 0
\(463\) 0.771316 + 0.308788i 0.771316 + 0.308788i 0.723734 0.690079i \(-0.242424\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.580057 0.814576i \(-0.696970\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(468\) 0 0
\(469\) 0.428368 + 1.23769i 0.428368 + 1.23769i
\(470\) 0 0
\(471\) 0.481929 + 0.676774i 0.481929 + 0.676774i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.56199 + 0.625325i 1.56199 + 0.625325i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(480\) 0 0
\(481\) 0.900739 + 3.71290i 0.900739 + 3.71290i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −0.642315 + 0.123796i −0.642315 + 0.123796i −0.500000 0.866025i \(-0.666667\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(488\) 0 0
\(489\) 0.581419 + 0.299742i 0.581419 + 0.299742i
\(490\) 0 0
\(491\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.959493 1.66189i 0.959493 1.66189i 0.235759 0.971812i \(-0.424242\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.48193 2.56678i −1.48193 2.56678i
\(508\) 0 0
\(509\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(510\) 0 0
\(511\) −1.97562 0.580093i −1.97562 0.580093i
\(512\) 0 0
\(513\) 0.0800569 + 1.68060i 0.0800569 + 1.68060i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(522\) 0 0
\(523\) 0.601300 0.573338i 0.601300 0.573338i −0.327068 0.945001i \(-0.606061\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(524\) 0 0
\(525\) −1.21590 + 0.486774i −1.21590 + 0.486774i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.995472 + 0.0950560i −0.995472 + 0.0950560i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.0930932 0.647478i 0.0930932 0.647478i −0.888835 0.458227i \(-0.848485\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(542\) 0 0
\(543\) −0.0947329 + 0.00904590i −0.0947329 + 0.00904590i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −0.607279 + 0.243118i −0.607279 + 0.243118i −0.654861 0.755750i \(-0.727273\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(548\) 0 0
\(549\) −1.28656 + 1.22673i −1.28656 + 1.22673i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0.620049 + 1.79151i 0.620049 + 1.79151i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.786053 0.618159i \(-0.212121\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(558\) 0 0
\(559\) −0.0446683 0.937702i −0.0446683 0.937702i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.947890 0.903811i −0.947890 0.903811i
\(568\) 0 0
\(569\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(570\) 0 0
\(571\) −0.0845850 + 1.77566i −0.0845850 + 1.77566i 0.415415 + 0.909632i \(0.363636\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.54370 0.297523i −1.54370 0.297523i −0.654861 0.755750i \(-0.727273\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(578\) 0 0
\(579\) −0.759713 + 0.876756i −0.759713 + 0.876756i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(588\) 0 0
\(589\) −2.04577 2.36094i −2.04577 2.36094i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.82318 + 0.729892i 1.82318 + 0.729892i
\(598\) 0 0
\(599\) 0 0 0.327068 0.945001i \(-0.393939\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(600\) 0 0
\(601\) 1.07701 + 1.51245i 1.07701 + 1.51245i 0.841254 + 0.540641i \(0.181818\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(602\) 0 0
\(603\) 0.981929 + 0.189251i 0.981929 + 0.189251i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.379436 + 1.09631i −0.379436 + 1.09631i 0.580057 + 0.814576i \(0.303030\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.74555 + 0.899892i −1.74555 + 0.899892i −0.786053 + 0.618159i \(0.787879\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(618\) 0 0
\(619\) −1.45949 1.14776i −1.45949 1.14776i −0.959493 0.281733i \(-0.909091\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.841254 + 0.540641i 0.841254 + 0.540641i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.396666 1.63508i 0.396666 1.63508i −0.327068 0.945001i \(-0.606061\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(632\) 0 0
\(633\) 0.786053 1.36148i 0.786053 1.36148i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.826153 + 1.16017i −0.826153 + 1.16017i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 0.186393 + 1.29639i 0.186393 + 1.29639i 0.841254 + 0.540641i \(0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2.42080 + 0.231158i 2.42080 + 0.231158i
\(652\) 0 0
\(653\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.13779 + 1.08488i −1.13779 + 1.08488i
\(658\) 0 0
\(659\) 0 0 0.928368 0.371662i \(-0.121212\pi\)
−0.928368 + 0.371662i \(0.878788\pi\)
\(660\) 0 0
\(661\) −0.271738 + 0.595023i −0.271738 + 0.595023i −0.995472 0.0950560i \(-0.969697\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0.0951638 0.0951638
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.56199 1.00383i 1.56199 1.00383i 0.580057 0.814576i \(-0.303030\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(674\) 0 0
\(675\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(676\) 0 0
\(677\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(678\) 0 0
\(679\) −0.0517765 + 0.113375i −0.0517765 + 0.113375i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.723734 0.690079i \(-0.242424\pi\)
−0.723734 + 0.690079i \(0.757576\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −0.473420 1.36786i −0.473420 1.36786i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.50842 1.18624i 1.50842 1.18624i 0.580057 0.814576i \(-0.303030\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(702\) 0 0
\(703\) −1.87283 + 2.63003i −1.87283 + 2.63003i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.396666 1.63508i 0.396666 1.63508i −0.327068 0.945001i \(-0.606061\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(710\) 0 0
\(711\) 1.42131 + 0.273935i 1.42131 + 0.273935i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.981929 0.189251i \(-0.0606061\pi\)
−0.981929 + 0.189251i \(0.939394\pi\)
\(720\) 0 0
\(721\) 0.855348 + 0.672653i 0.855348 + 0.672653i
\(722\) 0 0
\(723\) −0.947890 1.09392i −0.947890 1.09392i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.74555 + 0.899892i −1.74555 + 0.899892i −0.786053 + 0.618159i \(0.787879\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(728\) 0 0
\(729\) −0.959493 + 0.281733i −0.959493 + 0.281733i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −0.271738 + 0.785135i −0.271738 + 0.785135i 0.723734 + 0.690079i \(0.242424\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.13915 + 1.59971i 1.13915 + 1.59971i 0.723734 + 0.690079i \(0.242424\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(740\) 0 0
\(741\) 1.09560 3.16554i 1.09560 3.16554i
\(742\) 0 0
\(743\) 0 0 −0.928368 0.371662i \(-0.878788\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.857685 + 0.989821i 0.857685 + 0.989821i 1.00000 \(0\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.65033 0.850806i −1.65033 0.850806i −0.995472 0.0950560i \(-0.969697\pi\)
−0.654861 0.755750i \(-0.727273\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(762\) 0 0
\(763\) 2.02181 + 0.389672i 2.02181 + 0.389672i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.0224357 0.470984i 0.0224357 0.470984i −0.959493 0.281733i \(-0.909091\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.723734 0.690079i \(-0.757576\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(774\) 0 0
\(775\) −0.928368 1.60798i −0.928368 1.60798i
\(776\) 0 0
\(777\) −0.357685 2.48775i −0.357685 2.48775i
\(778\) 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
\(787\) 0.627639 + 1.81344i 0.627639 + 1.81344i 0.580057 + 0.814576i \(0.303030\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.28572 1.31540i 3.28572 1.31540i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.995472 0.0950560i \(-0.0303030\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(810\) 0 0
\(811\) −1.44091 + 0.137591i −1.44091 + 0.137591i −0.786053 0.618159i \(-0.787879\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(812\) 0 0
\(813\) 0.481929 1.05528i 0.481929 1.05528i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.574161 0.547462i 0.574161 0.547462i
\(818\) 0 0
\(819\) 1.08323 + 2.37194i 1.08323 + 2.37194i
\(820\) 0 0
\(821\) 0 0 −0.327068 0.945001i \(-0.606061\pi\)
0.327068 + 0.945001i \(0.393939\pi\)
\(822\) 0 0
\(823\) −1.15486 0.110276i −1.15486 0.110276i −0.500000 0.866025i \(-0.666667\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.0475819 0.998867i \(-0.515152\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(828\) 0 0
\(829\) 0.627639 + 0.184291i 0.627639 + 0.184291i 0.580057 0.814576i \(-0.303030\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(830\) 0 0
\(831\) 0.0930932 + 0.647478i 0.0930932 + 0.647478i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.07701 1.51245i 1.07701 1.51245i
\(838\) 0 0
\(839\) 0 0 0.0475819 0.998867i \(-0.484848\pi\)
−0.0475819 + 0.998867i \(0.515152\pi\)
\(840\) 0 0
\(841\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0.857685 0.989821i 0.857685 0.989821i
\(848\) 0 0
\(849\) −0.239446 0.153882i −0.239446 0.153882i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.462997 0.0892353i 0.462997 0.0892353i 0.0475819 0.998867i \(-0.484848\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(858\) 0 0
\(859\) 0.111165 + 0.458227i 0.111165 + 0.458227i 1.00000 \(0\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.327068 + 0.945001i −0.327068 + 0.945001i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1.67489 1.07639i −1.67489 1.07639i
\(872\) 0 0
\(873\) 0.0552004 + 0.0775182i 0.0552004 + 0.0775182i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.84833 0.739959i −1.84833 0.739959i −0.959493 0.281733i \(-0.909091\pi\)
−0.888835 0.458227i \(-0.848485\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.888835 0.458227i \(-0.151515\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(882\) 0 0
\(883\) 0.396666 + 1.63508i 0.396666 + 1.63508i 0.723734 + 0.690079i \(0.242424\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.786053 0.618159i \(-0.787879\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(888\) 0 0
\(889\) −0.606397 + 0.116873i −0.606397 + 0.116873i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −0.0293845 + 0.616858i −0.0293845 + 0.616858i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.205996 0.196417i −0.205996 0.196417i 0.580057 0.814576i \(-0.303030\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.283341 + 0.0270558i 0.283341 + 0.0270558i 0.235759 0.971812i \(-0.424242\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(920\) 0 0
\(921\) −0.642315 1.85585i −0.642315 1.85585i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.38884 + 1.32425i −1.38884 + 1.32425i
\(926\) 0 0
\(927\) 0.771316 0.308788i 0.771316 0.308788i
\(928\) 0 0
\(929\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(930\) 0 0
\(931\) −1.19817 + 0.114411i −1.19817 + 0.114411i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.96386 1.96386 0.981929 0.189251i \(-0.0606061\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(938\) 0 0
\(939\) 0.830830 0.830830
\(940\) 0 0
\(941\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(948\) 0 0
\(949\) 2.90577 1.16329i 2.90577 1.16329i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.116455 + 2.44470i 0.116455 + 2.44470i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.0475819 0.0824143i −0.0475819 0.0824143i 0.841254 0.540641i \(-0.181818\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.580057 0.814576i \(-0.303030\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(972\) 0 0
\(973\) −0.00593052 + 0.124497i −0.00593052 + 0.124497i
\(974\) 0 0
\(975\) 0.995472 1.72421i 0.995472 1.72421i
\(976\) 0 0
\(977\) 0 0 0.235759 0.971812i \(-0.424242\pi\)
−0.235759 + 0.971812i \(0.575758\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.02951 1.18812i 1.02951 1.18812i
\(982\) 0 0
\(983\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.186393 + 0.215109i 0.186393 + 0.215109i 0.841254 0.540641i \(-0.181818\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(992\) 0 0
\(993\) −0.419102 1.72756i −0.419102 1.72756i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.61435 + 0.474017i −1.61435 + 0.474017i −0.959493 0.281733i \(-0.909091\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(998\) 0 0
\(999\) −1.78153 0.713215i −1.78153 0.713215i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 804.1.be.a.629.1 yes 20
3.2 odd 2 CM 804.1.be.a.629.1 yes 20
4.3 odd 2 3216.1.dh.a.3041.1 20
12.11 even 2 3216.1.dh.a.3041.1 20
67.49 even 33 inner 804.1.be.a.317.1 20
201.116 odd 66 inner 804.1.be.a.317.1 20
268.183 odd 66 3216.1.dh.a.1121.1 20
804.719 even 66 3216.1.dh.a.1121.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
804.1.be.a.317.1 20 67.49 even 33 inner
804.1.be.a.317.1 20 201.116 odd 66 inner
804.1.be.a.629.1 yes 20 1.1 even 1 trivial
804.1.be.a.629.1 yes 20 3.2 odd 2 CM
3216.1.dh.a.1121.1 20 268.183 odd 66
3216.1.dh.a.1121.1 20 804.719 even 66
3216.1.dh.a.3041.1 20 4.3 odd 2
3216.1.dh.a.3041.1 20 12.11 even 2