Properties

Label 1012.1.r.a.945.1
Level $1012$
Weight $1$
Character 1012.945
Analytic conductor $0.505$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1012,1,Mod(197,1012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1012, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 14]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1012.197");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1012 = 2^{2} \cdot 11 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1012.r (of order \(22\), degree \(10\), 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: \(0.505053792785\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
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: \(\Z[a_1, \ldots, a_{23}]\)
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 945.1
Root \(0.723734 - 0.690079i\) of defining polynomial
Character \(\chi\) \(=\) 1012.945
Dual form 1012.1.r.a.725.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.279486 - 1.94387i) q^{3} +(1.70566 + 0.500828i) q^{5} +(-2.74102 + 0.804835i) q^{9} +O(q^{10})\) \(q+(-0.279486 - 1.94387i) q^{3} +(1.70566 + 0.500828i) q^{5} +(-2.74102 + 0.804835i) q^{9} +(0.841254 - 0.540641i) q^{11} +(0.496834 - 3.45556i) q^{15} +(-0.888835 - 0.458227i) q^{23} +(1.81720 + 1.16785i) q^{25} +(1.51475 + 3.31685i) q^{27} +(-0.0671040 + 0.466718i) q^{31} +(-1.28605 - 1.48418i) q^{33} +(-1.11312 + 0.326842i) q^{37} -5.07834 q^{45} -1.91899 q^{47} +(-0.142315 - 0.989821i) q^{49} +(-0.544078 + 0.627899i) q^{53} +(1.70566 - 0.500828i) q^{55} +(1.30379 + 1.50465i) q^{59} +(-0.550294 - 0.353653i) q^{67} +(-0.642315 + 1.85585i) q^{69} +(0.0800569 + 0.0514495i) q^{71} +(1.76226 - 3.85880i) q^{75} +(3.62093 - 2.32703i) q^{81} +(0.0930932 + 0.647478i) q^{89} +0.925994 q^{93} +(-0.452418 - 0.132842i) q^{97} +(-1.87076 + 2.15898i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 2 q^{5} - 2 q^{11} - 13 q^{15} + q^{23} - 2 q^{27} + 2 q^{31} - 9 q^{33} + 2 q^{37} - 4 q^{47} - 2 q^{49} - 4 q^{53} + 2 q^{55} + 2 q^{59} + 2 q^{67} - 12 q^{69} - 9 q^{71} + 22 q^{75} + 2 q^{81} + 2 q^{89} - 2 q^{93} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1012\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(507\) \(925\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{11}\right)\)

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 0
\(3\) −0.279486 1.94387i −0.279486 1.94387i −0.327068 0.945001i \(-0.606061\pi\)
0.0475819 0.998867i \(-0.484848\pi\)
\(4\) 0 0
\(5\) 1.70566 + 0.500828i 1.70566 + 0.500828i 0.981929 0.189251i \(-0.0606061\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(6\) 0 0
\(7\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(8\) 0 0
\(9\) −2.74102 + 0.804835i −2.74102 + 0.804835i
\(10\) 0 0
\(11\) 0.841254 0.540641i 0.841254 0.540641i
\(12\) 0 0
\(13\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(14\) 0 0
\(15\) 0.496834 3.45556i 0.496834 3.45556i
\(16\) 0 0
\(17\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(18\) 0 0
\(19\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.888835 0.458227i −0.888835 0.458227i
\(24\) 0 0
\(25\) 1.81720 + 1.16785i 1.81720 + 1.16785i
\(26\) 0 0
\(27\) 1.51475 + 3.31685i 1.51475 + 3.31685i
\(28\) 0 0
\(29\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(30\) 0 0
\(31\) −0.0671040 + 0.466718i −0.0671040 + 0.466718i 0.928368 + 0.371662i \(0.121212\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(32\) 0 0
\(33\) −1.28605 1.48418i −1.28605 1.48418i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.11312 + 0.326842i −1.11312 + 0.326842i −0.786053 0.618159i \(-0.787879\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(42\) 0 0
\(43\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(44\) 0 0
\(45\) −5.07834 −5.07834
\(46\) 0 0
\(47\) −1.91899 −1.91899 −0.959493 0.281733i \(-0.909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(48\) 0 0
\(49\) −0.142315 0.989821i −0.142315 0.989821i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.544078 + 0.627899i −0.544078 + 0.627899i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(54\) 0 0
\(55\) 1.70566 0.500828i 1.70566 0.500828i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.30379 + 1.50465i 1.30379 + 1.50465i 0.723734 + 0.690079i \(0.242424\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(60\) 0 0
\(61\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.550294 0.353653i −0.550294 0.353653i 0.235759 0.971812i \(-0.424242\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(68\) 0 0
\(69\) −0.642315 + 1.85585i −0.642315 + 1.85585i
\(70\) 0 0
\(71\) 0.0800569 + 0.0514495i 0.0800569 + 0.0514495i 0.580057 0.814576i \(-0.303030\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(72\) 0 0
\(73\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(74\) 0 0
\(75\) 1.76226 3.85880i 1.76226 3.85880i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(80\) 0 0
\(81\) 3.62093 2.32703i 3.62093 2.32703i
\(82\) 0 0
\(83\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.0930932 + 0.647478i 0.0930932 + 0.647478i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.925994 0.925994
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.452418 0.132842i −0.452418 0.132842i 0.0475819 0.998867i \(-0.484848\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0 0
\(99\) −1.87076 + 2.15898i −1.87076 + 2.15898i
\(100\) 0 0
\(101\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(102\) 0 0
\(103\) −0.239446 + 0.153882i −0.239446 + 0.153882i −0.654861 0.755750i \(-0.727273\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(108\) 0 0
\(109\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(110\) 0 0
\(111\) 0.946439 + 2.07241i 0.946439 + 2.07241i
\(112\) 0 0
\(113\) 1.21769 + 0.782560i 1.21769 + 0.782560i 0.981929 0.189251i \(-0.0606061\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(114\) 0 0
\(115\) −1.28656 1.22673i −1.28656 1.22673i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.415415 0.909632i 0.415415 0.909632i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.35052 + 1.55858i 1.35052 + 1.55858i
\(126\) 0 0
\(127\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.922490 + 6.41606i 0.922490 + 6.41606i
\(136\) 0 0
\(137\) 1.85674 1.85674 0.928368 0.371662i \(-0.121212\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0.536330 + 3.73026i 0.536330 + 3.73026i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.88431 + 0.553283i −1.88431 + 0.553283i
\(148\) 0 0
\(149\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(150\) 0 0
\(151\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.348202 + 0.762457i −0.348202 + 0.762457i
\(156\) 0 0
\(157\) −0.415415 0.909632i −0.415415 0.909632i −0.995472 0.0950560i \(-0.969697\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(158\) 0 0
\(159\) 1.37262 + 0.882127i 1.37262 + 0.882127i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(164\) 0 0
\(165\) −1.45025 3.17561i −1.45025 3.17561i
\(166\) 0 0
\(167\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(168\) 0 0
\(169\) −0.142315 + 0.989821i −0.142315 + 0.989821i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.56046 2.95493i 2.56046 2.95493i
\(178\) 0 0
\(179\) 1.50842 + 0.442913i 1.50842 + 0.442913i 0.928368 0.371662i \(-0.121212\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(180\) 0 0
\(181\) −0.264241 1.83784i −0.264241 1.83784i −0.500000 0.866025i \(-0.666667\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.06230 −2.06230
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.28605 + 1.48418i −1.28605 + 1.48418i −0.500000 + 0.866025i \(0.666667\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(192\) 0 0
\(193\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(198\) 0 0
\(199\) 0.186393 1.29639i 0.186393 1.29639i −0.654861 0.755750i \(-0.727273\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(200\) 0 0
\(201\) −0.533654 + 1.16854i −0.533654 + 1.16854i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.80511 + 0.540641i 2.80511 + 0.540641i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(212\) 0 0
\(213\) 0.0776362 0.169999i 0.0776362 0.169999i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.02951 1.18812i 1.02951 1.18812i 0.0475819 0.998867i \(-0.484848\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(224\) 0 0
\(225\) −5.92091 1.73854i −5.92091 1.73854i
\(226\) 0 0
\(227\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(228\) 0 0
\(229\) 1.44747 1.44747 0.723734 0.690079i \(-0.242424\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(234\) 0 0
\(235\) −3.27314 0.961081i −3.27314 0.961081i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(240\) 0 0
\(241\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(242\) 0 0
\(243\) −3.14757 3.63249i −3.14757 3.63249i
\(244\) 0 0
\(245\) 0.252989 1.75958i 0.252989 1.75958i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.67489 1.07639i −1.67489 1.07639i −0.888835 0.458227i \(-0.848485\pi\)
−0.786053 0.618159i \(-0.787879\pi\)
\(252\) 0 0
\(253\) −0.995472 + 0.0950560i −0.995472 + 0.0950560i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.345139 + 0.755750i 0.345139 + 0.755750i 1.00000 \(0\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(264\) 0 0
\(265\) −1.24248 + 0.798495i −1.24248 + 0.798495i
\(266\) 0 0
\(267\) 1.23259 0.361922i 1.23259 0.361922i
\(268\) 0 0
\(269\) −1.10181 + 1.27155i −1.10181 + 1.27155i −0.142315 + 0.989821i \(0.545455\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(270\) 0 0
\(271\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.16011 2.16011
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −0.191698 1.33329i −0.191698 1.33329i
\(280\) 0 0
\(281\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(282\) 0 0
\(283\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.654861 0.755750i −0.654861 0.755750i
\(290\) 0 0
\(291\) −0.131783 + 0.916569i −0.131783 + 0.916569i
\(292\) 0 0
\(293\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(294\) 0 0
\(295\) 1.47025 + 3.21941i 1.47025 + 3.21941i
\(296\) 0 0
\(297\) 3.06752 + 1.97137i 3.06752 + 1.97137i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(308\) 0 0
\(309\) 0.366049 + 0.422443i 0.366049 + 0.422443i
\(310\) 0 0
\(311\) 0.698939 0.449181i 0.698939 0.449181i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(312\) 0 0
\(313\) −0.0913090 + 0.0268107i −0.0913090 + 0.0268107i −0.327068 0.945001i \(-0.606061\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.38884 0.407799i −1.38884 0.407799i −0.500000 0.866025i \(-0.666667\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −0.0913090 + 0.0268107i −0.0913090 + 0.0268107i −0.327068 0.945001i \(-0.606061\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(332\) 0 0
\(333\) 2.78803 1.79176i 2.78803 1.79176i
\(334\) 0 0
\(335\) −0.761497 0.878815i −0.761497 0.878815i
\(336\) 0 0
\(337\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(338\) 0 0
\(339\) 1.18087 2.58574i 1.18087 2.58574i
\(340\) 0 0
\(341\) 0.195876 + 0.428908i 0.195876 + 0.428908i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2.02503 + 2.84376i −2.02503 + 2.84376i
\(346\) 0 0
\(347\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(348\) 0 0
\(349\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.223734 1.55610i 0.223734 1.55610i −0.500000 0.866025i \(-0.666667\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(354\) 0 0
\(355\) 0.110783 + 0.127850i 0.110783 + 0.127850i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(360\) 0 0
\(361\) −0.654861 + 0.755750i −0.654861 + 0.755750i
\(362\) 0 0
\(363\) −1.88431 0.553283i −1.88431 0.553283i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.0951638 0.0951638 0.0475819 0.998867i \(-0.484848\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(374\) 0 0
\(375\) 2.65223 3.06083i 2.65223 3.06083i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.67489 + 1.07639i −1.67489 + 1.07639i −0.786053 + 0.618159i \(0.787879\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −0.264241 + 1.83784i −0.264241 + 1.83784i 0.235759 + 0.971812i \(0.424242\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.32254 0.849945i −1.32254 0.849945i −0.327068 0.945001i \(-0.606061\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.698939 1.53046i 0.698939 1.53046i −0.142315 0.989821i \(-0.545455\pi\)
0.841254 0.540641i \(-0.181818\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.25667 + 1.45027i 1.25667 + 1.45027i 0.841254 + 0.540641i \(0.181818\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 7.34152 2.15566i 7.34152 2.15566i
\(406\) 0 0
\(407\) −0.759713 + 0.876756i −0.759713 + 0.876756i
\(408\) 0 0
\(409\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(410\) 0 0
\(411\) −0.518932 3.60925i −0.518932 3.60925i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.273100 + 0.0801894i 0.273100 + 0.0801894i 0.415415 0.909632i \(-0.363636\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(420\) 0 0
\(421\) −1.10181 + 1.27155i −1.10181 + 1.27155i −0.142315 + 0.989821i \(0.545455\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(422\) 0 0
\(423\) 5.25998 1.54447i 5.25998 1.54447i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(432\) 0 0
\(433\) −0.653077 1.43004i −0.653077 1.43004i −0.888835 0.458227i \(-0.848485\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(440\) 0 0
\(441\) 1.18673 + 2.59858i 1.18673 + 2.59858i
\(442\) 0 0
\(443\) −0.738471 + 1.61703i −0.738471 + 1.61703i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(444\) 0 0
\(445\) −0.165489 + 1.15100i −0.165489 + 1.15100i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.396666 0.254922i 0.396666 0.254922i −0.327068 0.945001i \(-0.606061\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −0.205996 1.43273i −0.205996 1.43273i −0.786053 0.618159i \(-0.787879\pi\)
0.580057 0.814576i \(-0.303030\pi\)
\(464\) 0 0
\(465\) 1.57943 + 0.463763i 1.57943 + 0.463763i
\(466\) 0 0
\(467\) −0.947890 + 1.09392i −0.947890 + 1.09392i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1.65210 + 1.06174i −1.65210 + 1.06174i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.985972 2.15898i 0.985972 2.15898i
\(478\) 0 0
\(479\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.705142 0.453167i −0.705142 0.453167i
\(486\) 0 0
\(487\) −0.271738 0.595023i −0.271738 0.595023i 0.723734 0.690079i \(-0.242424\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(488\) 0 0
\(489\) −0.232205 + 0.508459i −0.232205 + 0.508459i
\(490\) 0 0
\(491\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.27217 + 2.74556i −4.27217 + 2.74556i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.857685 0.989821i 0.857685 0.989821i −0.142315 0.989821i \(-0.545455\pi\)
1.00000 \(0\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.96386 1.96386
\(508\) 0 0
\(509\) 0.0930932 + 0.647478i 0.0930932 + 0.647478i 0.981929 + 0.189251i \(0.0606061\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.485482 + 0.142550i −0.485482 + 0.142550i
\(516\) 0 0
\(517\) −1.61435 + 1.03748i −1.61435 + 1.03748i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.283341 1.97068i 0.283341 1.97068i 0.0475819 0.998867i \(-0.484848\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(522\) 0 0
\(523\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.580057 + 0.814576i 0.580057 + 0.814576i
\(530\) 0 0
\(531\) −4.78471 3.07495i −4.78471 3.07495i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.439382 3.05597i 0.439382 3.05597i
\(538\) 0 0
\(539\) −0.654861 0.755750i −0.654861 0.755750i
\(540\) 0 0
\(541\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(542\) 0 0
\(543\) −3.49866 + 1.02730i −3.49866 + 1.02730i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.576384 + 4.00884i 0.576384 + 4.00884i
\(556\) 0 0
\(557\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(564\) 0 0
\(565\) 1.68504 + 1.94464i 1.68504 + 1.94464i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(570\) 0 0
\(571\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(572\) 0 0
\(573\) 3.24449 + 2.08511i 3.24449 + 2.08511i
\(574\) 0 0
\(575\) −1.08006 1.87071i −1.08006 1.87071i
\(576\) 0 0
\(577\) −1.67489 1.07639i −1.67489 1.07639i −0.888835 0.458227i \(-0.848485\pi\)
−0.786053 0.618159i \(-0.787879\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.118239 + 0.822373i −0.118239 + 0.822373i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.10181 + 0.708089i −1.10181 + 0.708089i −0.959493 0.281733i \(-0.909091\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.57211 −2.57211
\(598\) 0 0
\(599\) −1.30972 −1.30972 −0.654861 0.755750i \(-0.727273\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(600\) 0 0
\(601\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(602\) 0 0
\(603\) 1.79300 + 0.526472i 1.79300 + 0.526472i
\(604\) 0 0
\(605\) 1.16413 1.34347i 1.16413 1.34347i
\(606\) 0 0
\(607\) 0 0 0.959493 0.281733i \(-0.0909091\pi\)
−0.959493 + 0.281733i \(0.909091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.544078 1.19136i −0.544078 1.19136i −0.959493 0.281733i \(-0.909091\pi\)
0.415415 0.909632i \(-0.363636\pi\)
\(618\) 0 0
\(619\) 0.0800569 + 0.0514495i 0.0800569 + 0.0514495i 0.580057 0.814576i \(-0.303030\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) 0.173501 3.64223i 0.173501 3.64223i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.625606 + 1.36989i 0.625606 + 1.36989i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.308779 0.356349i −0.308779 0.356349i 0.580057 0.814576i \(-0.303030\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.260846 0.0765912i −0.260846 0.0765912i
\(640\) 0 0
\(641\) −0.165101 1.14831i −0.165101 1.14831i −0.888835 0.458227i \(-0.848485\pi\)
0.723734 0.690079i \(-0.242424\pi\)
\(642\) 0 0
\(643\) 0.471518 0.471518 0.235759 0.971812i \(-0.424242\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.264241 1.83784i −0.264241 1.83784i −0.500000 0.866025i \(-0.666667\pi\)
0.235759 0.971812i \(-0.424242\pi\)
\(648\) 0 0
\(649\) 1.91030 + 0.560914i 1.91030 + 0.560914i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.50842 0.442913i 1.50842 0.442913i 0.580057 0.814576i \(-0.303030\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(660\) 0 0
\(661\) −0.653077 + 1.43004i −0.653077 + 1.43004i 0.235759 + 0.971812i \(0.424242\pi\)
−0.888835 + 0.458227i \(0.848485\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −2.59728 1.66917i −2.59728 1.66917i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(674\) 0 0
\(675\) −1.12095 + 7.79639i −1.12095 + 7.79639i
\(676\) 0 0
\(677\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.186393 0.215109i 0.186393 0.215109i −0.654861 0.755750i \(-0.727273\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(684\) 0 0
\(685\) 3.16697 + 0.929905i 3.16697 + 0.929905i
\(686\) 0 0
\(687\) −0.404547 2.81369i −0.404547 2.81369i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.85674 1.85674 0.928368 0.371662i \(-0.121212\pi\)
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 0.841254 0.540641i \(-0.181818\pi\)
−0.841254 + 0.540641i \(0.818182\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −0.953418 + 6.63117i −0.953418 + 6.63117i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.815816 + 1.78639i 0.815816 + 1.78639i 0.580057 + 0.814576i \(0.303030\pi\)
0.235759 + 0.971812i \(0.424242\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.273507 0.384087i 0.273507 0.384087i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −0.271738 + 0.595023i −0.271738 + 0.595023i −0.995472 0.0950560i \(-0.969697\pi\)
0.723734 + 0.690079i \(0.242424\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.91030 0.560914i 1.91030 0.560914i 0.928368 0.371662i \(-0.121212\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(728\) 0 0
\(729\) −3.36273 + 3.88080i −3.36273 + 3.88080i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(734\) 0 0
\(735\) −3.49109 −3.49109
\(736\) 0 0
\(737\) −0.654136 −0.654136
\(738\) 0 0
\(739\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.142315 0.989821i 0.142315 0.989821i −0.786053 0.618159i \(-0.787879\pi\)
0.928368 0.371662i \(-0.121212\pi\)
\(752\) 0 0
\(753\) −1.62424 + 3.55660i −1.62424 + 3.55660i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.239446 0.153882i −0.239446 0.153882i 0.415415 0.909632i \(-0.363636\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(758\) 0 0
\(759\) 0.462997 + 1.90850i 0.462997 + 1.90850i
\(760\) 0 0
\(761\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(770\) 0 0
\(771\) 1.37262 0.882127i 1.37262 0.882127i
\(772\) 0 0
\(773\) 1.84125 0.540641i 1.84125 0.540641i 0.841254 0.540641i \(-0.181818\pi\)
1.00000 \(0\)
\(774\) 0 0
\(775\) −0.666997 + 0.769755i −0.666997 + 0.769755i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0.0951638 0.0951638
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.252989 1.75958i −0.252989 1.75958i
\(786\) 0 0
\(787\) 0 0 −0.959493 0.281733i \(-0.909091\pi\)
0.959493 + 0.281733i \(0.0909091\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 1.89943 + 2.19205i 1.89943 + 2.19205i
\(796\) 0 0
\(797\) −0.279486 + 1.94387i −0.279486 + 1.94387i 0.0475819 + 0.998867i \(0.484848\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −0.776283 1.69982i −0.776283 1.69982i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.77967 + 1.78639i 2.77967 + 1.78639i
\(808\) 0 0
\(809\) 0 0 −0.415415 0.909632i \(-0.636364\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(810\) 0 0
\(811\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −0.331345 0.382393i −0.331345 0.382393i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.654861 0.755750i \(-0.272727\pi\)
−0.654861 + 0.755750i \(0.727273\pi\)
\(822\) 0 0
\(823\) 1.91030 + 0.560914i 1.91030 + 0.560914i 0.981929 + 0.189251i \(0.0606061\pi\)
0.928368 + 0.371662i \(0.121212\pi\)
\(824\) 0 0
\(825\) −0.603722 4.19898i −0.603722 4.19898i
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.57211 −1.57211 −0.786053 0.618159i \(-0.787879\pi\)
−0.786053 + 0.618159i \(0.787879\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.64968 + 0.484390i −1.64968 + 0.484390i
\(838\) 0 0
\(839\) 1.21769 0.782560i 1.21769 0.782560i 0.235759 0.971812i \(-0.424242\pi\)
0.981929 + 0.189251i \(0.0606061\pi\)
\(840\) 0 0
\(841\) −0.654861 0.755750i −0.654861 0.755750i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.738471 + 1.61703i −0.738471 + 1.61703i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.13915 + 0.219553i 1.13915 + 0.219553i
\(852\) 0 0
\(853\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(858\) 0 0
\(859\) 0.0930932 0.647478i 0.0930932 0.647478i −0.888835 0.458227i \(-0.848485\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.41542 0.909632i 1.41542 0.909632i 0.415415 0.909632i \(-0.363636\pi\)
1.00000 \(0\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.28605 + 1.48418i −1.28605 + 1.48418i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.34700 1.34700
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.142315 0.989821i \(-0.545455\pi\)
0.142315 + 0.989821i \(0.454545\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.02951 1.18812i 1.02951 1.18812i 0.0475819 0.998867i \(-0.484848\pi\)
0.981929 0.189251i \(-0.0606061\pi\)
\(882\) 0 0
\(883\) 1.84125 0.540641i 1.84125 0.540641i 0.841254 0.540641i \(-0.181818\pi\)
1.00000 \(0\)
\(884\) 0 0
\(885\) 5.84719 3.75776i 5.84719 3.75776i
\(886\) 0 0
\(887\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.78803 3.91524i 1.78803 3.91524i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 2.35104 + 1.51092i 2.35104 + 1.51092i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.469734 3.26707i 0.469734 3.26707i
\(906\) 0 0
\(907\) −1.10181 1.27155i −1.10181 1.27155i −0.959493 0.281733i \(-0.909091\pi\)
−0.142315 0.989821i \(-0.545455\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.273100 0.0801894i 0.273100 0.0801894i −0.142315 0.989821i \(-0.545455\pi\)
0.415415 + 0.909632i \(0.363636\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.40447 0.706016i −2.40447 0.706016i
\(926\) 0 0
\(927\) 0.532475 0.614509i 0.532475 0.614509i
\(928\) 0 0
\(929\) 1.25667 0.368991i 1.25667 0.368991i 0.415415 0.909632i \(-0.363636\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −1.06849 1.23310i −1.06849 1.23310i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.415415 0.909632i \(-0.363636\pi\)
−0.415415 + 0.909632i \(0.636364\pi\)
\(938\) 0 0
\(939\) 0.0776362 + 0.169999i 0.0776362 + 0.169999i
\(940\) 0 0
\(941\) 0 0 −0.841254 0.540641i \(-0.818182\pi\)
0.841254 + 0.540641i \(0.181818\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.771316 + 1.68895i 0.771316 + 1.68895i 0.723734 + 0.690079i \(0.242424\pi\)
0.0475819 + 0.998867i \(0.484848\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −0.404547 + 2.81369i −0.404547 + 2.81369i
\(952\) 0 0
\(953\) 0 0 −0.654861 0.755750i \(-0.727273\pi\)
0.654861 + 0.755750i \(0.272727\pi\)
\(954\) 0 0
\(955\) −2.93689 + 1.88743i −2.93689 + 1.88743i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.746170 + 0.219095i 0.746170 + 0.219095i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.78153 0.523103i −1.78153 0.523103i −0.786053 0.618159i \(-0.787879\pi\)
−0.995472 + 0.0950560i \(0.969697\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.32254 + 0.849945i −1.32254 + 0.849945i −0.995472 0.0950560i \(-0.969697\pi\)
−0.327068 + 0.945001i \(0.606061\pi\)
\(978\) 0 0
\(979\) 0.428368 + 0.494363i 0.428368 + 0.494363i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −0.827068 + 1.81103i −0.827068 + 1.81103i −0.327068 + 0.945001i \(0.606061\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.41542 + 0.909632i 1.41542 + 0.909632i 1.00000 \(0\)
0.415415 + 0.909632i \(0.363636\pi\)
\(992\) 0 0
\(993\) 0.0776362 + 0.169999i 0.0776362 + 0.169999i
\(994\) 0 0
\(995\) 0.967192 2.11785i 0.967192 2.11785i
\(996\) 0 0
\(997\) 0 0 0.142315 0.989821i \(-0.454545\pi\)
−0.142315 + 0.989821i \(0.545455\pi\)
\(998\) 0 0
\(999\) −2.77019 3.19697i −2.77019 3.19697i
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 1012.1.r.a.945.1 yes 20
11.10 odd 2 CM 1012.1.r.a.945.1 yes 20
23.12 even 11 inner 1012.1.r.a.725.1 20
253.219 odd 22 inner 1012.1.r.a.725.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1012.1.r.a.725.1 20 23.12 even 11 inner
1012.1.r.a.725.1 20 253.219 odd 22 inner
1012.1.r.a.945.1 yes 20 1.1 even 1 trivial
1012.1.r.a.945.1 yes 20 11.10 odd 2 CM