Properties

Label 1944.1.bh.a.355.1
Level $1944$
Weight $1$
Character 1944.355
Analytic conductor $0.970$
Analytic rank $0$
Dimension $54$
Projective image $D_{81}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1944,1,Mod(43,1944)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1944, base_ring=CyclotomicField(162))
 
chi = DirichletCharacter(H, H._module([81, 81, 130]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1944.43");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1944.bh (of order \(162\), degree \(54\), 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: \(0.970182384559\)
Analytic rank: \(0\)
Dimension: \(54\)
Coefficient field: \(\Q(\zeta_{162})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{54} - x^{27} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{81}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{81} + \cdots)\)

Embedding invariants

Embedding label 355.1
Root \(0.431386 + 0.902167i\) of defining polynomial
Character \(\chi\) \(=\) 1944.355
Dual form 1944.1.bh.a.115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.135331 - 0.990800i) q^{2} +(-0.360178 - 0.932884i) q^{3} +(-0.963371 + 0.268173i) q^{4} +(-0.875558 + 0.483113i) q^{6} +(0.396080 + 0.918216i) q^{8} +(-0.740544 + 0.672008i) q^{9} +O(q^{10})\) \(q+(-0.135331 - 0.990800i) q^{2} +(-0.360178 - 0.932884i) q^{3} +(-0.963371 + 0.268173i) q^{4} +(-0.875558 + 0.483113i) q^{6} +(0.396080 + 0.918216i) q^{8} +(-0.740544 + 0.672008i) q^{9} +(0.938848 + 1.07580i) q^{11} +(0.597159 + 0.802123i) q^{12} +(0.856167 - 0.516699i) q^{16} +(0.906963 + 0.214954i) q^{17} +(0.766044 + 0.642788i) q^{18} +(1.34676 + 1.42748i) q^{19} +(0.938848 - 1.07580i) q^{22} +(0.713930 - 0.700217i) q^{24} +(-0.999248 - 0.0387754i) q^{25} +(0.893633 + 0.448799i) q^{27} +(-0.627812 - 0.778365i) q^{32} +(0.665445 - 1.26332i) q^{33} +(0.0902362 - 0.927709i) q^{34} +(0.533204 - 0.845986i) q^{36} +(1.23209 - 1.52755i) q^{38} +(-1.56180 - 1.21049i) q^{41} +(0.569364 + 1.47469i) q^{43} +(-1.19296 - 0.784622i) q^{44} +(-0.790393 - 0.612601i) q^{48} +(0.987990 - 0.154519i) q^{49} +(0.0968109 + 0.995303i) q^{50} +(-0.126141 - 0.923512i) q^{51} +(0.323734 - 0.946148i) q^{54} +(0.846600 - 1.77051i) q^{57} +(-1.68024 - 0.329988i) q^{59} +(-0.686242 + 0.727374i) q^{64} +(-1.34175 - 0.488357i) q^{66} +(1.89877 + 0.608816i) q^{67} +(-0.931386 + 0.0361420i) q^{68} +(-0.910363 - 0.413811i) q^{72} +(0.420545 + 0.0491546i) q^{73} +(0.323734 + 0.946148i) q^{75} +(-1.68024 - 1.01403i) q^{76} +(0.0968109 - 0.995303i) q^{81} +(-0.987990 + 1.71125i) q^{82} +(0.0919145 - 0.0712391i) q^{83} +(1.38407 - 0.763697i) q^{86} +(-0.615959 + 1.28817i) q^{88} +(-0.819590 + 1.10090i) q^{89} +(-0.500000 + 0.866025i) q^{96} +(-0.0353063 - 1.82038i) q^{97} +(-0.286803 - 0.957990i) q^{98} +(-1.41820 - 0.165764i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 54 q+O(q^{10}) \) Copy content Toggle raw display \( 54 q - 27 q^{68} - 27 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(973\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{37}{81}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.135331 0.990800i −0.135331 0.990800i
\(3\) −0.360178 0.932884i −0.360178 0.932884i
\(4\) −0.963371 + 0.268173i −0.963371 + 0.268173i
\(5\) 0 0 0.0193913 0.999812i \(-0.493827\pi\)
−0.0193913 + 0.999812i \(0.506173\pi\)
\(6\) −0.875558 + 0.483113i −0.875558 + 0.483113i
\(7\) 0 0 0.996993 0.0774924i \(-0.0246914\pi\)
−0.996993 + 0.0774924i \(0.975309\pi\)
\(8\) 0.396080 + 0.918216i 0.396080 + 0.918216i
\(9\) −0.740544 + 0.672008i −0.740544 + 0.672008i
\(10\) 0 0
\(11\) 0.938848 + 1.07580i 0.938848 + 1.07580i 0.996993 + 0.0774924i \(0.0246914\pi\)
−0.0581448 + 0.998308i \(0.518519\pi\)
\(12\) 0.597159 + 0.802123i 0.597159 + 0.802123i
\(13\) 0 0 0.466044 0.884762i \(-0.345679\pi\)
−0.466044 + 0.884762i \(0.654321\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.856167 0.516699i 0.856167 0.516699i
\(17\) 0.906963 + 0.214954i 0.906963 + 0.214954i 0.657521 0.753436i \(-0.271605\pi\)
0.249441 + 0.968390i \(0.419753\pi\)
\(18\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(19\) 1.34676 + 1.42748i 1.34676 + 1.42748i 0.813552 + 0.581492i \(0.197531\pi\)
0.533204 + 0.845986i \(0.320988\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.938848 1.07580i 0.938848 1.07580i
\(23\) 0 0 0.565607 0.824675i \(-0.308642\pi\)
−0.565607 + 0.824675i \(0.691358\pi\)
\(24\) 0.713930 0.700217i 0.713930 0.700217i
\(25\) −0.999248 0.0387754i −0.999248 0.0387754i
\(26\) 0 0
\(27\) 0.893633 + 0.448799i 0.893633 + 0.448799i
\(28\) 0 0
\(29\) 0 0 0.952248 0.305326i \(-0.0987654\pi\)
−0.952248 + 0.305326i \(0.901235\pi\)
\(30\) 0 0
\(31\) 0 0 −0.910363 0.413811i \(-0.864198\pi\)
0.910363 + 0.413811i \(0.135802\pi\)
\(32\) −0.627812 0.778365i −0.627812 0.778365i
\(33\) 0.665445 1.26332i 0.665445 1.26332i
\(34\) 0.0902362 0.927709i 0.0902362 0.927709i
\(35\) 0 0
\(36\) 0.533204 0.845986i 0.533204 0.845986i
\(37\) 0 0 0.893633 0.448799i \(-0.148148\pi\)
−0.893633 + 0.448799i \(0.851852\pi\)
\(38\) 1.23209 1.52755i 1.23209 1.52755i
\(39\) 0 0
\(40\) 0 0
\(41\) −1.56180 1.21049i −1.56180 1.21049i −0.875558 0.483113i \(-0.839506\pi\)
−0.686242 0.727374i \(-0.740741\pi\)
\(42\) 0 0
\(43\) 0.569364 + 1.47469i 0.569364 + 1.47469i 0.856167 + 0.516699i \(0.172840\pi\)
−0.286803 + 0.957990i \(0.592593\pi\)
\(44\) −1.19296 0.784622i −1.19296 0.784622i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.910363 0.413811i \(-0.135802\pi\)
−0.910363 + 0.413811i \(0.864198\pi\)
\(48\) −0.790393 0.612601i −0.790393 0.612601i
\(49\) 0.987990 0.154519i 0.987990 0.154519i
\(50\) 0.0968109 + 0.995303i 0.0968109 + 0.995303i
\(51\) −0.126141 0.923512i −0.126141 0.923512i
\(52\) 0 0
\(53\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(54\) 0.323734 0.946148i 0.323734 0.946148i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.846600 1.77051i 0.846600 1.77051i
\(58\) 0 0
\(59\) −1.68024 0.329988i −1.68024 0.329988i −0.740544 0.672008i \(-0.765432\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(60\) 0 0
\(61\) 0 0 −0.963371 0.268173i \(-0.913580\pi\)
0.963371 + 0.268173i \(0.0864198\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.686242 + 0.727374i −0.686242 + 0.727374i
\(65\) 0 0
\(66\) −1.34175 0.488357i −1.34175 0.488357i
\(67\) 1.89877 + 0.608816i 1.89877 + 0.608816i 0.973045 + 0.230616i \(0.0740741\pi\)
0.925724 + 0.378200i \(0.123457\pi\)
\(68\) −0.931386 + 0.0361420i −0.931386 + 0.0361420i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.597159 0.802123i \(-0.703704\pi\)
0.597159 + 0.802123i \(0.296296\pi\)
\(72\) −0.910363 0.413811i −0.910363 0.413811i
\(73\) 0.420545 + 0.0491546i 0.420545 + 0.0491546i 0.323734 0.946148i \(-0.395062\pi\)
0.0968109 + 0.995303i \(0.469136\pi\)
\(74\) 0 0
\(75\) 0.323734 + 0.946148i 0.323734 + 0.946148i
\(76\) −1.68024 1.01403i −1.68024 1.01403i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.925724 0.378200i \(-0.876543\pi\)
0.925724 + 0.378200i \(0.123457\pi\)
\(80\) 0 0
\(81\) 0.0968109 0.995303i 0.0968109 0.995303i
\(82\) −0.987990 + 1.71125i −0.987990 + 1.71125i
\(83\) 0.0919145 0.0712391i 0.0919145 0.0712391i −0.565607 0.824675i \(-0.691358\pi\)
0.657521 + 0.753436i \(0.271605\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.38407 0.763697i 1.38407 0.763697i
\(87\) 0 0
\(88\) −0.615959 + 1.28817i −0.615959 + 1.28817i
\(89\) −0.819590 + 1.10090i −0.819590 + 1.10090i 0.173648 + 0.984808i \(0.444444\pi\)
−0.993238 + 0.116093i \(0.962963\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(97\) −0.0353063 1.82038i −0.0353063 1.82038i −0.431386 0.902167i \(-0.641975\pi\)
0.396080 0.918216i \(-0.370370\pi\)
\(98\) −0.286803 0.957990i −0.286803 0.957990i
\(99\) −1.41820 0.165764i −1.41820 0.165764i
\(100\) 0.973045 0.230616i 0.973045 0.230616i
\(101\) 0 0 0.713930 0.700217i \(-0.246914\pi\)
−0.713930 + 0.700217i \(0.753086\pi\)
\(102\) −0.897946 + 0.249960i −0.897946 + 0.249960i
\(103\) 0 0 −0.323734 0.946148i \(-0.604938\pi\)
0.323734 + 0.946148i \(0.395062\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.257188 1.45859i −0.257188 1.45859i −0.790393 0.612601i \(-0.790123\pi\)
0.533204 0.845986i \(-0.320988\pi\)
\(108\) −0.981255 0.192712i −0.981255 0.192712i
\(109\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.285318 + 0.738993i −0.285318 + 0.738993i 0.713930 + 0.700217i \(0.246914\pi\)
−0.999248 + 0.0387754i \(0.987654\pi\)
\(114\) −1.86880 0.599206i −1.86880 0.599206i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −0.0995634 + 1.70944i −0.0995634 + 1.70944i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.140580 + 1.02923i −0.140580 + 1.02923i
\(122\) 0 0
\(123\) −0.566717 + 1.89297i −0.566717 + 1.89297i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 0.835488 0.549509i \(-0.185185\pi\)
−0.835488 + 0.549509i \(0.814815\pi\)
\(128\) 0.813552 + 0.581492i 0.813552 + 0.581492i
\(129\) 1.17064 1.06230i 1.17064 1.06230i
\(130\) 0 0
\(131\) 0.971639 0.694486i 0.971639 0.694486i 0.0193913 0.999812i \(-0.493827\pi\)
0.952248 + 0.305326i \(0.0987654\pi\)
\(132\) −0.302283 + 1.39550i −0.302283 + 1.39550i
\(133\) 0 0
\(134\) 0.346252 1.96369i 0.346252 1.96369i
\(135\) 0 0
\(136\) 0.161855 + 0.917927i 0.161855 + 0.917927i
\(137\) 0.867579 1.37651i 0.867579 1.37651i −0.0581448 0.998308i \(-0.518519\pi\)
0.925724 0.378200i \(-0.123457\pi\)
\(138\) 0 0
\(139\) 0.193040 + 0.0150042i 0.193040 + 0.0150042i 0.173648 0.984808i \(-0.444444\pi\)
0.0193913 + 0.999812i \(0.493827\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.286803 + 0.957990i −0.286803 + 0.957990i
\(145\) 0 0
\(146\) −0.00821044 0.423328i −0.00821044 0.423328i
\(147\) −0.500000 0.866025i −0.500000 0.866025i
\(148\) 0 0
\(149\) 0 0 −0.533204 0.845986i \(-0.679012\pi\)
0.533204 + 0.845986i \(0.320988\pi\)
\(150\) 0.893633 0.448799i 0.893633 0.448799i
\(151\) 0 0 0.323734 0.946148i \(-0.395062\pi\)
−0.323734 + 0.946148i \(0.604938\pi\)
\(152\) −0.777311 + 1.80201i −0.777311 + 1.80201i
\(153\) −0.816096 + 0.450303i −0.816096 + 0.450303i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.875558 0.483113i \(-0.160494\pi\)
−0.875558 + 0.483113i \(0.839506\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.999248 + 0.0387754i −0.999248 + 0.0387754i
\(163\) 0.993238 + 1.72034i 0.993238 + 1.72034i 0.597159 + 0.802123i \(0.296296\pi\)
0.396080 + 0.918216i \(0.370370\pi\)
\(164\) 1.82921 + 0.747315i 1.82921 + 0.747315i
\(165\) 0 0
\(166\) −0.0830226 0.0814280i −0.0830226 0.0814280i
\(167\) 0 0 −0.856167 0.516699i \(-0.827160\pi\)
0.856167 + 0.516699i \(0.172840\pi\)
\(168\) 0 0
\(169\) −0.565607 0.824675i −0.565607 0.824675i
\(170\) 0 0
\(171\) −1.95661 0.152080i −1.95661 0.152080i
\(172\) −0.943980 1.26798i −0.943980 1.26798i
\(173\) 0 0 0.981255 0.192712i \(-0.0617284\pi\)
−0.981255 + 0.192712i \(0.938272\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.35968 + 0.435963i 1.35968 + 0.435963i
\(177\) 0.297344 + 1.68632i 0.297344 + 1.68632i
\(178\) 1.20169 + 0.663064i 1.20169 + 0.663064i
\(179\) −1.33549 + 1.41553i −1.33549 + 1.41553i −0.500000 + 0.866025i \(0.666667\pi\)
−0.835488 + 0.549509i \(0.814815\pi\)
\(180\) 0 0
\(181\) 0 0 0.286803 0.957990i \(-0.407407\pi\)
−0.286803 + 0.957990i \(0.592593\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.620252 + 1.17752i 0.620252 + 1.17752i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.211704 0.977334i \(-0.432099\pi\)
−0.211704 + 0.977334i \(0.567901\pi\)
\(192\) 0.925724 + 0.378200i 0.925724 + 0.378200i
\(193\) −0.0697382 0.716972i −0.0697382 0.716972i −0.963371 0.268173i \(-0.913580\pi\)
0.893633 0.448799i \(-0.148148\pi\)
\(194\) −1.79886 + 0.281336i −1.79886 + 0.281336i
\(195\) 0 0
\(196\) −0.910363 + 0.413811i −0.910363 + 0.413811i
\(197\) 0 0 −0.893633 0.448799i \(-0.851852\pi\)
0.893633 + 0.448799i \(0.148148\pi\)
\(198\) 0.0276881 + 1.42759i 0.0276881 + 1.42759i
\(199\) 0 0 −0.835488 0.549509i \(-0.814815\pi\)
0.835488 + 0.549509i \(0.185185\pi\)
\(200\) −0.360178 0.932884i −0.360178 0.932884i
\(201\) −0.115940 1.99061i −0.115940 1.99061i
\(202\) 0 0
\(203\) 0 0
\(204\) 0.369181 + 0.855857i 0.369181 + 0.855857i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.271282 + 2.78903i −0.271282 + 2.78903i
\(210\) 0 0
\(211\) −1.12207 1.39114i −1.12207 1.39114i −0.910363 0.413811i \(-0.864198\pi\)
−0.211704 0.977334i \(-0.567901\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −1.41036 + 0.452215i −1.41036 + 0.452215i
\(215\) 0 0
\(216\) −0.0581448 + 0.998308i −0.0581448 + 0.998308i
\(217\) 0 0
\(218\) 0 0
\(219\) −0.105615 0.410024i −0.105615 0.410024i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.249441 0.968390i \(-0.580247\pi\)
0.249441 + 0.968390i \(0.419753\pi\)
\(224\) 0 0
\(225\) 0.766044 0.642788i 0.766044 0.642788i
\(226\) 0.770807 + 0.182685i 0.770807 + 0.182685i
\(227\) 1.58515 0.956642i 1.58515 0.956642i 0.597159 0.802123i \(-0.296296\pi\)
0.987990 0.154519i \(-0.0493827\pi\)
\(228\) −0.340786 + 1.93270i −0.340786 + 1.93270i
\(229\) 0 0 −0.211704 0.977334i \(-0.567901\pi\)
0.211704 + 0.977334i \(0.432099\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.73928 0.203292i 1.73928 0.203292i 0.813552 0.581492i \(-0.197531\pi\)
0.925724 + 0.378200i \(0.123457\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.70718 0.132693i 1.70718 0.132693i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.963371 0.268173i \(-0.0864198\pi\)
−0.963371 + 0.268173i \(0.913580\pi\)
\(240\) 0 0
\(241\) 0.116760 + 0.854835i 0.116760 + 0.854835i 0.952248 + 0.305326i \(0.0987654\pi\)
−0.835488 + 0.549509i \(0.814815\pi\)
\(242\) 1.03878 1.03878
\(243\) −0.963371 + 0.268173i −0.963371 + 0.268173i
\(244\) 0 0
\(245\) 0 0
\(246\) 1.95225 + 0.305326i 1.95225 + 0.305326i
\(247\) 0 0
\(248\) 0 0
\(249\) −0.0995634 0.0600868i −0.0995634 0.0600868i
\(250\) 0 0
\(251\) −0.107204 0.248527i −0.107204 0.248527i 0.856167 0.516699i \(-0.172840\pi\)
−0.963371 + 0.268173i \(0.913580\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.466044 0.884762i 0.466044 0.884762i
\(257\) 0.182652 + 0.843216i 0.182652 + 0.843216i 0.973045 + 0.230616i \(0.0740741\pi\)
−0.790393 + 0.612601i \(0.790123\pi\)
\(258\) −1.21095 1.01611i −1.21095 1.01611i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −0.819590 0.868715i −0.819590 0.868715i
\(263\) 0 0 −0.249441 0.968390i \(-0.580247\pi\)
0.249441 + 0.968390i \(0.419753\pi\)
\(264\) 1.42357 + 0.110648i 1.42357 + 0.110648i
\(265\) 0 0
\(266\) 0 0
\(267\) 1.32221 + 0.368062i 1.32221 + 0.368062i
\(268\) −1.99249 0.0773175i −1.99249 0.0773175i
\(269\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(270\) 0 0
\(271\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(272\) 0.887578 0.284590i 0.887578 0.284590i
\(273\) 0 0
\(274\) −1.48126 0.673313i −1.48126 0.673313i
\(275\) −0.896427 1.11140i −0.896427 1.11140i
\(276\) 0 0
\(277\) 0 0 0.0968109 0.995303i \(-0.469136\pi\)
−0.0968109 + 0.995303i \(0.530864\pi\)
\(278\) −0.0112581 0.193294i −0.0112581 0.193294i
\(279\) 0 0
\(280\) 0 0
\(281\) −1.12207 + 1.39114i −1.12207 + 1.39114i −0.211704 + 0.977334i \(0.567901\pi\)
−0.910363 + 0.413811i \(0.864198\pi\)
\(282\) 0 0
\(283\) −1.54686 + 0.631963i −1.54686 + 0.631963i −0.981255 0.192712i \(-0.938272\pi\)
−0.565607 + 0.824675i \(0.691358\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.987990 + 0.154519i 0.987990 + 0.154519i
\(289\) −0.117257 0.0588886i −0.117257 0.0588886i
\(290\) 0 0
\(291\) −1.68549 + 0.688598i −1.68549 + 0.688598i
\(292\) −0.418323 + 0.0654245i −0.418323 + 0.0654245i
\(293\) 0 0 −0.0968109 0.995303i \(-0.530864\pi\)
0.0968109 + 0.995303i \(0.469136\pi\)
\(294\) −0.790393 + 0.612601i −0.790393 + 0.612601i
\(295\) 0 0
\(296\) 0 0
\(297\) 0.356167 + 1.38272i 0.356167 + 1.38272i
\(298\) 0 0
\(299\) 0 0
\(300\) −0.565607 0.824675i −0.565607 0.824675i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.89063 + 0.526292i 1.89063 + 0.526292i
\(305\) 0 0
\(306\) 0.556604 + 0.747649i 0.556604 + 0.747649i
\(307\) 1.20169 1.27372i 1.20169 1.27372i 0.249441 0.968390i \(-0.419753\pi\)
0.952248 0.305326i \(-0.0987654\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.999248 0.0387754i \(-0.0123457\pi\)
−0.999248 + 0.0387754i \(0.987654\pi\)
\(312\) 0 0
\(313\) 1.23209 0.241974i 1.23209 0.241974i 0.466044 0.884762i \(-0.345679\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.565607 0.824675i \(-0.691358\pi\)
0.565607 + 0.824675i \(0.308642\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −1.26806 + 0.765277i −1.26806 + 0.765277i
\(322\) 0 0
\(323\) 0.914615 + 1.58416i 0.914615 + 1.58416i
\(324\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(325\) 0 0
\(326\) 1.57010 1.21692i 1.57010 1.21692i
\(327\) 0 0
\(328\) 0.492891 1.91352i 0.492891 1.91352i
\(329\) 0 0
\(330\) 0 0
\(331\) −0.839516 + 1.75570i −0.839516 + 1.75570i −0.211704 + 0.977334i \(0.567901\pi\)
−0.627812 + 0.778365i \(0.716049\pi\)
\(332\) −0.0694434 + 0.0932786i −0.0694434 + 0.0932786i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.144318 0.228977i −0.144318 0.228977i 0.766044 0.642788i \(-0.222222\pi\)
−0.910363 + 0.413811i \(0.864198\pi\)
\(338\) −0.740544 + 0.672008i −0.740544 + 0.672008i
\(339\) 0.792160 0.792160
\(340\) 0 0
\(341\) 0 0
\(342\) 0.114110 + 1.95919i 0.114110 + 1.95919i
\(343\) 0 0
\(344\) −1.12857 + 1.10689i −1.12857 + 1.10689i
\(345\) 0 0
\(346\) 0 0
\(347\) 1.31109 + 0.101906i 1.31109 + 0.101906i 0.713930 0.700217i \(-0.246914\pi\)
0.597159 + 0.802123i \(0.296296\pi\)
\(348\) 0 0
\(349\) 0 0 0.533204 0.845986i \(-0.320988\pi\)
−0.533204 + 0.845986i \(0.679012\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.247945 1.40617i 0.247945 1.40617i
\(353\) −1.41036 1.27984i −1.41036 1.27984i −0.910363 0.413811i \(-0.864198\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(354\) 1.63057 0.522820i 1.63057 0.522820i
\(355\) 0 0
\(356\) 0.494338 1.28037i 0.494338 1.28037i
\(357\) 0 0
\(358\) 1.58325 + 1.13164i 1.58325 + 1.13164i
\(359\) 0 0 0.835488 0.549509i \(-0.185185\pi\)
−0.835488 + 0.549509i \(0.814815\pi\)
\(360\) 0 0
\(361\) −0.165797 + 2.84662i −0.165797 + 2.84662i
\(362\) 0 0
\(363\) 1.01078 0.239560i 1.01078 0.239560i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.987990 0.154519i \(-0.950617\pi\)
0.987990 + 0.154519i \(0.0493827\pi\)
\(368\) 0 0
\(369\) 1.97004 0.153123i 1.97004 0.153123i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.360178 0.932884i \(-0.382716\pi\)
−0.360178 + 0.932884i \(0.617284\pi\)
\(374\) 1.08275 0.773902i 1.08275 0.773902i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.218037 1.23655i −0.218037 1.23655i −0.875558 0.483113i \(-0.839506\pi\)
0.657521 0.753436i \(-0.271605\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.323734 0.946148i \(-0.604938\pi\)
0.323734 + 0.946148i \(0.395062\pi\)
\(384\) 0.249441 0.968390i 0.249441 0.968390i
\(385\) 0 0
\(386\) −0.700938 + 0.166125i −0.700938 + 0.166125i
\(387\) −1.41264 0.709455i −1.41264 0.709455i
\(388\) 0.522190 + 1.74424i 0.522190 + 1.74424i
\(389\) 0 0 −0.0193913 0.999812i \(-0.506173\pi\)
0.0193913 + 0.999812i \(0.493827\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.533204 + 0.845986i 0.533204 + 0.845986i
\(393\) −0.997837 0.656288i −0.997837 0.656288i
\(394\) 0 0
\(395\) 0 0
\(396\) 1.41071 0.220631i 1.41071 0.220631i
\(397\) 0 0 0.597159 0.802123i \(-0.296296\pi\)
−0.597159 + 0.802123i \(0.703704\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.875558 + 0.483113i −0.875558 + 0.483113i
\(401\) 0.161505 0.627001i 0.161505 0.627001i −0.835488 0.549509i \(-0.814815\pi\)
0.996993 0.0774924i \(-0.0246914\pi\)
\(402\) −1.95661 + 0.384266i −1.95661 + 0.384266i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0.798022 0.481609i 0.798022 0.481609i
\(409\) 0.761341 + 0.746718i 0.761341 + 0.746718i 0.973045 0.230616i \(-0.0740741\pi\)
−0.211704 + 0.977334i \(0.567901\pi\)
\(410\) 0 0
\(411\) −1.59660 0.313563i −1.59660 0.313563i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.0555313 0.185488i −0.0555313 0.185488i
\(418\) 2.80008 0.108656i 2.80008 0.108656i
\(419\) −1.89162 0.606523i −1.89162 0.606523i −0.981255 0.192712i \(-0.938272\pi\)
−0.910363 0.413811i \(-0.864198\pi\)
\(420\) 0 0
\(421\) 0 0 −0.875558 0.483113i \(-0.839506\pi\)
0.875558 + 0.483113i \(0.160494\pi\)
\(422\) −1.22650 + 1.30001i −1.22650 + 1.30001i
\(423\) 0 0
\(424\) 0 0
\(425\) −0.897946 0.249960i −0.897946 0.249960i
\(426\) 0 0
\(427\) 0 0
\(428\) 0.638921 + 1.33619i 0.638921 + 1.33619i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(432\) 0.996993 0.0774924i 0.996993 0.0774924i
\(433\) −1.53094 1.28461i −1.53094 1.28461i −0.790393 0.612601i \(-0.790123\pi\)
−0.740544 0.672008i \(-0.765432\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −0.391959 + 0.160133i −0.391959 + 0.160133i
\(439\) 0 0 0.910363 0.413811i \(-0.135802\pi\)
−0.910363 + 0.413811i \(0.864198\pi\)
\(440\) 0 0
\(441\) −0.627812 + 0.778365i −0.627812 + 0.778365i
\(442\) 0 0
\(443\) 0.259456 + 0.672008i 0.259456 + 0.672008i 1.00000 \(0\)
−0.740544 + 0.672008i \(0.765432\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.173027 0.0868973i 0.173027 0.0868973i −0.360178 0.932884i \(-0.617284\pi\)
0.533204 + 0.845986i \(0.320988\pi\)
\(450\) −0.740544 0.672008i −0.740544 0.672008i
\(451\) −0.164051 2.81665i −0.164051 2.81665i
\(452\) 0.0766897 0.788439i 0.0766897 0.788439i
\(453\) 0 0
\(454\) −1.16236 1.44110i −1.16236 1.44110i
\(455\) 0 0
\(456\) 1.96103 + 0.0760971i 1.96103 + 0.0760971i
\(457\) −1.66750 + 0.534661i −1.66750 + 0.534661i −0.981255 0.192712i \(-0.938272\pi\)
−0.686242 + 0.727374i \(0.740741\pi\)
\(458\) 0 0
\(459\) 0.714020 + 0.599134i 0.714020 + 0.599134i
\(460\) 0 0
\(461\) 0 0 −0.999248 0.0387754i \(-0.987654\pi\)
0.999248 + 0.0387754i \(0.0123457\pi\)
\(462\) 0 0
\(463\) 0 0 0.565607 0.824675i \(-0.308642\pi\)
−0.565607 + 0.824675i \(0.691358\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.436800 1.69576i −0.436800 1.69576i
\(467\) −0.444319 0.470951i −0.444319 0.470951i 0.466044 0.884762i \(-0.345679\pi\)
−0.910363 + 0.413811i \(0.864198\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −0.362508 1.67352i −0.362508 1.67352i
\(473\) −1.05193 + 1.99703i −1.05193 + 1.99703i
\(474\) 0 0
\(475\) −1.29039 1.47863i −1.29039 1.47863i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.996993 0.0774924i \(-0.0246914\pi\)
−0.996993 + 0.0774924i \(0.975309\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0.831170 0.231372i 0.831170 0.231372i
\(483\) 0 0
\(484\) −0.140580 1.02923i −0.140580 1.02923i
\(485\) 0 0
\(486\) 0.396080 + 0.918216i 0.396080 + 0.918216i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 1.24713 1.54620i 1.24713 1.54620i
\(490\) 0 0
\(491\) 0.0369307 1.90414i 0.0369307 1.90414i −0.286803 0.957990i \(-0.592593\pi\)
0.323734 0.946148i \(-0.395062\pi\)
\(492\) 0.0383169 1.97561i 0.0383169 1.97561i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −0.0460600 + 0.106779i −0.0460600 + 0.106779i
\(499\) −0.931386 + 1.76819i −0.931386 + 1.76819i −0.431386 + 0.902167i \(0.641975\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −0.231732 + 0.139851i −0.231732 + 0.139851i
\(503\) 0 0 −0.973045 0.230616i \(-0.925926\pi\)
0.973045 + 0.230616i \(0.0740741\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.565607 + 0.824675i −0.565607 + 0.824675i
\(508\) 0 0
\(509\) 0 0 0.565607 0.824675i \(-0.308642\pi\)
−0.565607 + 0.824675i \(0.691358\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.939693 0.342020i −0.939693 0.342020i
\(513\) 0.562854 + 1.88006i 0.562854 + 1.88006i
\(514\) 0.810741 0.295085i 0.810741 0.295085i
\(515\) 0 0
\(516\) −0.842882 + 1.33732i −0.842882 + 1.33732i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.107652 1.84832i −0.107652 1.84832i −0.431386 0.902167i \(-0.641975\pi\)
0.323734 0.946148i \(-0.395062\pi\)
\(522\) 0 0
\(523\) 0.707900 0.355521i 0.707900 0.355521i −0.0581448 0.998308i \(-0.518519\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(524\) −0.749807 + 0.929615i −0.749807 + 0.929615i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −0.0830226 1.42544i −0.0830226 1.42544i
\(529\) −0.360178 0.932884i −0.360178 0.932884i
\(530\) 0 0
\(531\) 1.46604 0.884762i 1.46604 0.884762i
\(532\) 0 0
\(533\) 0 0
\(534\) 0.185740 1.35986i 0.185740 1.35986i
\(535\) 0 0
\(536\) 0.193040 + 1.98462i 0.193040 + 1.98462i
\(537\) 1.80154 + 0.736011i 1.80154 + 0.736011i
\(538\) 0 0
\(539\) 1.09380 + 0.917810i 1.09380 + 0.917810i
\(540\) 0 0
\(541\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.402089 0.840899i −0.402089 0.840899i
\(545\) 0 0
\(546\) 0 0
\(547\) −1.02735 0.285982i −1.02735 0.285982i −0.286803 0.957990i \(-0.592593\pi\)
−0.740544 + 0.672008i \(0.765432\pi\)
\(548\) −0.466659 + 1.55875i −0.466659 + 1.55875i
\(549\) 0 0
\(550\) −0.979857 + 1.03859i −0.979857 + 1.03859i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −0.189992 + 0.0373133i −0.189992 + 0.0373133i
\(557\) 0 0 −0.597159 0.802123i \(-0.703704\pi\)
0.597159 + 0.802123i \(0.296296\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0.875088 1.00274i 0.875088 1.00274i
\(562\) 1.53020 + 0.923479i 1.53020 + 0.923479i
\(563\) 0.138232 + 0.135577i 0.138232 + 0.135577i 0.766044 0.642788i \(-0.222222\pi\)
−0.627812 + 0.778365i \(0.716049\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0.835488 + 1.44711i 0.835488 + 1.44711i
\(567\) 0 0
\(568\) 0 0
\(569\) −0.0306535 + 0.0237583i −0.0306535 + 0.0237583i −0.627812 0.778365i \(-0.716049\pi\)
0.597159 + 0.802123i \(0.296296\pi\)
\(570\) 0 0
\(571\) −0.369444 + 1.43427i −0.369444 + 1.43427i 0.466044 + 0.884762i \(0.345679\pi\)
−0.835488 + 0.549509i \(0.814815\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.0193913 0.999812i 0.0193913 0.999812i
\(577\) 0.644463 1.49403i 0.644463 1.49403i −0.211704 0.977334i \(-0.567901\pi\)
0.856167 0.516699i \(-0.172840\pi\)
\(578\) −0.0424783 + 0.124148i −0.0424783 + 0.124148i
\(579\) −0.643733 + 0.323295i −0.643733 + 0.323295i
\(580\) 0 0
\(581\) 0 0
\(582\) 0.910363 + 1.57679i 0.910363 + 1.57679i
\(583\) 0 0
\(584\) 0.121435 + 0.405620i 0.121435 + 0.405620i
\(585\) 0 0
\(586\) 0 0
\(587\) −0.615959 + 0.604128i −0.615959 + 0.604128i −0.939693 0.342020i \(-0.888889\pi\)
0.323734 + 0.946148i \(0.395062\pi\)
\(588\) 0.713930 + 0.700217i 0.713930 + 0.700217i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.0996057 0.564892i −0.0996057 0.564892i −0.993238 0.116093i \(-0.962963\pi\)
0.893633 0.448799i \(-0.148148\pi\)
\(594\) 1.32180 0.540016i 1.32180 0.540016i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.360178 0.932884i \(-0.382716\pi\)
−0.360178 + 0.932884i \(0.617284\pi\)
\(600\) −0.740544 + 0.672008i −0.740544 + 0.672008i
\(601\) −1.56750 1.12039i −1.56750 1.12039i −0.939693 0.342020i \(-0.888889\pi\)
−0.627812 0.778365i \(-0.716049\pi\)
\(602\) 0 0
\(603\) −1.81525 + 0.825133i −1.81525 + 0.825133i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.135331 0.990800i \(-0.456790\pi\)
−0.135331 + 0.990800i \(0.543210\pi\)
\(608\) 0.265589 1.94446i 0.265589 1.94446i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0.665445 0.652664i 0.665445 0.652664i
\(613\) 0 0 0.835488 0.549509i \(-0.185185\pi\)
−0.835488 + 0.549509i \(0.814815\pi\)
\(614\) −1.42462 1.01826i −1.42462 1.01826i
\(615\) 0 0
\(616\) 0 0
\(617\) −1.48126 + 1.05874i −1.48126 + 1.05874i −0.500000 + 0.866025i \(0.666667\pi\)
−0.981255 + 0.192712i \(0.938272\pi\)
\(618\) 0 0
\(619\) 0.533455 + 0.484085i 0.533455 + 0.484085i 0.893633 0.448799i \(-0.148148\pi\)
−0.360178 + 0.932884i \(0.617284\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.996993 + 0.0774924i 0.996993 + 0.0774924i
\(626\) −0.406488 1.18801i −0.406488 1.18801i
\(627\) 2.69955 0.751470i 2.69955 0.751470i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 −0.286803 0.957990i \(-0.592593\pi\)
0.286803 + 0.957990i \(0.407407\pi\)
\(632\) 0 0
\(633\) −0.893633 + 1.54782i −0.893633 + 1.54782i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.567291 + 1.18639i −0.567291 + 1.18639i 0.396080 + 0.918216i \(0.370370\pi\)
−0.963371 + 0.268173i \(0.913580\pi\)
\(642\) 0.929845 + 1.15283i 0.929845 + 1.15283i
\(643\) −0.436800 + 0.241016i −0.436800 + 0.241016i −0.686242 0.727374i \(-0.740741\pi\)
0.249441 + 0.968390i \(0.419753\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.44581 1.12059i 1.44581 1.12059i
\(647\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0.952248 0.305326i 0.952248 0.305326i
\(649\) −1.22249 2.11741i −1.22249 2.11741i
\(650\) 0 0
\(651\) 0 0
\(652\) −1.41820 1.39097i −1.41820 1.39097i
\(653\) 0 0 −0.856167 0.516699i \(-0.827160\pi\)
0.856167 + 0.516699i \(0.172840\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.96262 0.229397i −1.96262 0.229397i
\(657\) −0.344464 + 0.246208i −0.344464 + 0.246208i
\(658\) 0 0
\(659\) −1.75376 + 0.344428i −1.75376 + 0.344428i −0.963371 0.268173i \(-0.913580\pi\)
−0.790393 + 0.612601i \(0.790123\pi\)
\(660\) 0 0
\(661\) 0 0 0.999248 0.0387754i \(-0.0123457\pi\)
−0.999248 + 0.0387754i \(0.987654\pi\)
\(662\) 1.85316 + 0.594192i 1.85316 + 0.594192i
\(663\) 0 0
\(664\) 0.101818 + 0.0561810i 0.101818 + 0.0561810i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.778747 1.47842i −0.778747 1.47842i −0.875558 0.483113i \(-0.839506\pi\)
0.0968109 0.995303i \(-0.469136\pi\)
\(674\) −0.207340 + 0.173979i −0.207340 + 0.173979i
\(675\) −0.875558 0.483113i −0.875558 0.483113i
\(676\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(677\) 0 0 0.211704 0.977334i \(-0.432099\pi\)
−0.211704 + 0.977334i \(0.567901\pi\)
\(678\) −0.107204 0.784872i −0.107204 0.784872i
\(679\) 0 0
\(680\) 0 0
\(681\) −1.46337 1.13420i −1.46337 1.13420i
\(682\) 0 0
\(683\) −1.01089 0.507688i −1.01089 0.507688i −0.135331 0.990800i \(-0.543210\pi\)
−0.875558 + 0.483113i \(0.839506\pi\)
\(684\) 1.92572 0.378200i 1.92572 0.378200i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.24944 + 0.968390i 1.24944 + 0.968390i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.861662 1.06829i 0.861662 1.06829i −0.135331 0.990800i \(-0.543210\pi\)
0.996993 0.0774924i \(-0.0246914\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −0.0764629 1.31282i −0.0764629 1.31282i
\(695\) 0 0
\(696\) 0 0
\(697\) −1.15629 1.43358i −1.15629 1.43358i
\(698\) 0 0
\(699\) −0.816096 1.54932i −0.816096 1.54932i
\(700\) 0 0
\(701\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.42679 0.0553658i −1.42679 0.0553658i
\(705\) 0 0
\(706\) −1.07720 + 1.57059i −1.07720 + 1.57059i
\(707\) 0 0
\(708\) −0.738677 1.54481i −0.738677 1.54481i
\(709\) 0 0 −0.249441 0.968390i \(-0.580247\pi\)
0.249441 + 0.968390i \(0.419753\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.33549 0.316516i −1.33549 0.316516i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.906963 1.72183i 0.906963 1.72183i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.993238 0.116093i \(-0.0370370\pi\)
−0.993238 + 0.116093i \(0.962963\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.84287 0.220966i 2.84287 0.220966i
\(723\) 0.755407 0.416816i 0.755407 0.416816i
\(724\) 0 0
\(725\) 0 0
\(726\) −0.374146 0.969063i −0.374146 0.969063i
\(727\) 0 0 −0.135331 0.990800i \(-0.543210\pi\)
0.135331 + 0.990800i \(0.456790\pi\)
\(728\) 0 0
\(729\) 0.597159 + 0.802123i 0.597159 + 0.802123i
\(730\) 0 0
\(731\) 0.199401 + 1.45987i 0.199401 + 1.45987i
\(732\) 0 0
\(733\) 0 0 0.963371 0.268173i \(-0.0864198\pi\)
−0.963371 + 0.268173i \(0.913580\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.12769 + 2.61428i 1.12769 + 2.61428i
\(738\) −0.418323 1.93119i −0.418323 1.93119i
\(739\) 1.91371 0.223681i 1.91371 0.223681i 0.925724 0.378200i \(-0.123457\pi\)
0.987990 + 0.154519i \(0.0493827\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.211704 0.977334i \(-0.567901\pi\)
0.211704 + 0.977334i \(0.432099\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.0201935 + 0.114523i −0.0201935 + 0.114523i
\(748\) −0.913312 0.968054i −0.913312 0.968054i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.657521 0.753436i \(-0.271605\pi\)
−0.657521 + 0.753436i \(0.728395\pi\)
\(752\) 0 0
\(753\) −0.193234 + 0.189523i −0.193234 + 0.189523i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(758\) −1.19567 + 0.383375i −1.19567 + 0.383375i
\(759\) 0 0
\(760\) 0 0
\(761\) 1.17990 + 1.46285i 1.17990 + 1.46285i 0.856167 + 0.516699i \(0.172840\pi\)
0.323734 + 0.946148i \(0.395062\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.993238 0.116093i −0.993238 0.116093i
\(769\) 0.733321 0.299595i 0.733321 0.299595i 0.0193913 0.999812i \(-0.493827\pi\)
0.713930 + 0.700217i \(0.246914\pi\)
\(770\) 0 0
\(771\) 0.720836 0.474101i 0.720836 0.474101i
\(772\) 0.259456 + 0.672008i 0.259456 + 0.672008i
\(773\) 0 0 −0.835488 0.549509i \(-0.814815\pi\)
0.835488 + 0.549509i \(0.185185\pi\)
\(774\) −0.511754 + 1.49566i −0.511754 + 1.49566i
\(775\) 0 0
\(776\) 1.65752 0.753436i 1.65752 0.753436i
\(777\) 0 0
\(778\) 0 0
\(779\) −0.375421 3.85967i −0.375421 3.85967i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.766044 0.642788i 0.766044 0.642788i
\(785\) 0 0
\(786\) −0.515212 + 1.07747i −0.515212 + 1.07747i
\(787\) 0.0501657 + 0.104913i 0.0501657 + 0.104913i 0.925724 0.378200i \(-0.123457\pi\)
−0.875558 + 0.483113i \(0.839506\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.409515 1.36787i −0.409515 1.36787i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.999248 0.0387754i \(-0.0123457\pi\)
−0.999248 + 0.0387754i \(0.987654\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.597159 + 0.802123i 0.597159 + 0.802123i
\(801\) −0.132871 1.36604i −0.132871 1.36604i
\(802\) −0.643090 0.0751664i −0.643090 0.0751664i
\(803\) 0.341947 + 0.498571i 0.341947 + 0.498571i
\(804\) 0.645521 + 1.88661i 0.645521 + 1.88661i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.211704 + 0.366682i 0.211704 + 0.366682i 0.952248 0.305326i \(-0.0987654\pi\)
−0.740544 + 0.672008i \(0.765432\pi\)
\(810\) 0 0
\(811\) 0.565607 0.979660i 0.565607 0.979660i −0.431386 0.902167i \(-0.641975\pi\)
0.996993 0.0774924i \(-0.0246914\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −0.585176 0.725504i −0.585176 0.725504i
\(817\) −1.33829 + 2.79880i −1.33829 + 2.79880i
\(818\) 0.636815 0.855391i 0.636815 0.855391i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.323734 0.946148i \(-0.395062\pi\)
−0.323734 + 0.946148i \(0.604938\pi\)
\(822\) −0.0946077 + 1.62435i −0.0946077 + 1.62435i
\(823\) 0 0 −0.533204 0.845986i \(-0.679012\pi\)
0.533204 + 0.845986i \(0.320988\pi\)
\(824\) 0 0
\(825\) −0.713930 + 1.23656i −0.713930 + 1.23656i
\(826\) 0 0
\(827\) −0.439408 1.46773i −0.439408 1.46773i −0.835488 0.549509i \(-0.814815\pi\)
0.396080 0.918216i \(-0.370370\pi\)
\(828\) 0 0
\(829\) 0 0 0.973045 0.230616i \(-0.0740741\pi\)
−0.973045 + 0.230616i \(0.925926\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.929284 + 0.0722297i 0.929284 + 0.0722297i
\(834\) −0.176266 + 0.0801228i −0.176266 + 0.0801228i
\(835\) 0 0
\(836\) −0.486595 2.75962i −0.486595 2.75962i
\(837\) 0 0
\(838\) −0.344948 + 1.95630i −0.344948 + 1.95630i
\(839\) 0 0 −0.740544 0.672008i \(-0.765432\pi\)
0.740544 + 0.672008i \(0.234568\pi\)
\(840\) 0 0
\(841\) 0.813552 0.581492i 0.813552 0.581492i
\(842\) 0 0
\(843\) 1.70192 + 0.545699i 1.70192 + 0.545699i
\(844\) 1.45403 + 1.03928i 1.45403 + 1.03928i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.14669 + 1.21542i 1.14669 + 1.21542i
\(850\) −0.126141 + 0.923512i −0.126141 + 0.923512i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.987990 0.154519i \(-0.950617\pi\)
0.987990 + 0.154519i \(0.0493827\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.23743 0.813871i 1.23743 0.813871i
\(857\) 0.282544 + 0.201950i 0.282544 + 0.201950i 0.713930 0.700217i \(-0.246914\pi\)
−0.431386 + 0.902167i \(0.641975\pi\)
\(858\) 0 0
\(859\) 0.693969 1.79743i 0.693969 1.79743i 0.0968109 0.995303i \(-0.469136\pi\)
0.597159 0.802123i \(-0.296296\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(864\) −0.211704 0.977334i −0.211704 0.977334i
\(865\) 0 0
\(866\) −1.06561 + 1.69070i −1.06561 + 1.69070i
\(867\) −0.0127029 + 0.130597i −0.0127029 + 0.130597i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.24946 + 1.32435i 1.24946 + 1.32435i
\(874\) 0 0
\(875\) 0 0
\(876\) 0.211704 + 0.366682i 0.211704 + 0.366682i
\(877\) 0 0 0.740544 0.672008i \(-0.234568\pi\)
−0.740544 + 0.672008i \(0.765432\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.770807 1.78693i 0.770807 1.78693i 0.173648 0.984808i \(-0.444444\pi\)
0.597159 0.802123i \(-0.296296\pi\)
\(882\) 0.856167 + 0.516699i 0.856167 + 0.516699i
\(883\) 0.636815 0.855391i 0.636815 0.855391i −0.360178 0.932884i \(-0.617284\pi\)
0.996993 + 0.0774924i \(0.0246914\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.630713 0.348013i 0.630713 0.348013i
\(887\) 0 0 0.249441 0.968390i \(-0.419753\pi\)
−0.249441 + 0.968390i \(0.580247\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.16164 0.830289i 1.16164 0.830289i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.109514 0.159675i −0.109514 0.159675i
\(899\) 0 0
\(900\) −0.565607 + 0.824675i −0.565607 + 0.824675i
\(901\) 0 0
\(902\) −2.76853 + 0.543722i −2.76853 + 0.543722i
\(903\) 0 0
\(904\) −0.791564 + 0.0307163i −0.791564 + 0.0307163i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.755407 + 0.416816i 0.755407 + 0.416816i 0.813552 0.581492i \(-0.197531\pi\)
−0.0581448 + 0.998308i \(0.518519\pi\)
\(908\) −1.27054 + 1.34669i −1.27054 + 1.34669i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.963371 0.268173i \(-0.913580\pi\)
0.963371 + 0.268173i \(0.0864198\pi\)
\(912\) −0.189992 1.95329i −0.189992 1.95329i
\(913\) 0.162933 + 0.0319990i 0.162933 + 0.0319990i
\(914\) 0.755407 + 1.57980i 0.755407 + 1.57980i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0.496993 0.788533i 0.496993 0.788533i
\(919\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(920\) 0 0
\(921\) −1.62105 0.662272i −1.62105 0.662272i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −0.125088 0.323987i −0.125088 0.323987i 0.856167 0.516699i \(-0.172840\pi\)
−0.981255 + 0.192712i \(0.938272\pi\)
\(930\) 0 0
\(931\) 1.55115 + 1.20224i 1.55115 + 1.20224i
\(932\) −1.62105 + 0.662272i −1.62105 + 0.662272i
\(933\) 0 0
\(934\) −0.406488 + 0.503966i −0.406488 + 0.503966i
\(935\) 0 0
\(936\) 0 0
\(937\) 0.0333522 + 0.572636i 0.0333522 + 0.572636i 0.973045 + 0.230616i \(0.0740741\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(938\) 0 0
\(939\) −0.669504 1.06224i −0.669504 1.06224i
\(940\) 0 0
\(941\) 0 0 −0.910363 0.413811i \(-0.864198\pi\)
0.910363 + 0.413811i \(0.135802\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.60907 + 0.585653i −1.60907 + 0.585653i
\(945\) 0 0
\(946\) 2.12102 + 0.771987i 2.12102 + 0.771987i
\(947\) 0.423089 + 0.0164178i 0.423089 + 0.0164178i 0.249441 0.968390i \(-0.419753\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −1.29039 + 1.47863i −1.29039 + 1.47863i
\(951\) 0 0
\(952\) 0 0
\(953\) −1.36836 1.45037i −1.36836 1.45037i −0.740544 0.672008i \(-0.765432\pi\)
−0.627812 0.778365i \(-0.716049\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.657521 + 0.753436i 0.657521 + 0.753436i
\(962\) 0 0
\(963\) 1.17064 + 0.907315i 1.17064 + 0.907315i
\(964\) −0.341727 0.792211i −0.341727 0.792211i
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.0193913 0.999812i \(-0.493827\pi\)
−0.0193913 + 0.999812i \(0.506173\pi\)
\(968\) −1.00073 + 0.278573i −1.00073 + 0.278573i
\(969\) 1.14841 1.42381i 1.14841 1.42381i
\(970\) 0 0
\(971\) 1.53209 1.53209 0.766044 0.642788i \(-0.222222\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(972\) 0.856167 0.516699i 0.856167 0.516699i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.000752047 0.0387754i 0.000752047 0.0387754i −0.999248 0.0387754i \(-0.987654\pi\)
1.00000 \(0\)
\(978\) −1.70076 1.02641i −1.70076 1.02641i
\(979\) −1.95382 + 0.151863i −1.95382 + 0.151863i
\(980\) 0 0
\(981\) 0 0
\(982\) −1.89162 + 0.221098i −1.89162 + 0.221098i
\(983\) 0 0 −0.657521 0.753436i \(-0.728395\pi\)
0.657521 + 0.753436i \(0.271605\pi\)
\(984\) −1.96262 + 0.229397i −1.96262 + 0.229397i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.686242 0.727374i \(-0.740741\pi\)
0.686242 + 0.727374i \(0.259259\pi\)
\(992\) 0 0
\(993\) 1.94024 + 0.150807i 1.94024 + 0.150807i
\(994\) 0 0
\(995\) 0 0
\(996\) 0.112030 + 0.0311857i 0.112030 + 0.0311857i
\(997\) 0 0 −0.999248 0.0387754i \(-0.987654\pi\)
0.999248 + 0.0387754i \(0.0123457\pi\)
\(998\) 1.87797 + 0.683526i 1.87797 + 0.683526i
\(999\) 0 0
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 1944.1.bh.a.355.1 yes 54
8.3 odd 2 CM 1944.1.bh.a.355.1 yes 54
243.115 even 81 inner 1944.1.bh.a.115.1 54
1944.115 odd 162 inner 1944.1.bh.a.115.1 54
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1944.1.bh.a.115.1 54 243.115 even 81 inner
1944.1.bh.a.115.1 54 1944.115 odd 162 inner
1944.1.bh.a.355.1 yes 54 1.1 even 1 trivial
1944.1.bh.a.355.1 yes 54 8.3 odd 2 CM