Properties

Label 776.1.bp.a.99.1
Level $776$
Weight $1$
Character 776.99
Analytic conductor $0.387$
Analytic rank $0$
Dimension $16$
Projective image $D_{48}$
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] = [776,1,Mod(3,776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(776, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 24, 35]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("776.3");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 776 = 2^{3} \cdot 97 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 776.bp (of order \(48\), degree \(16\), 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: \(0.387274449803\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{48}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{48} + \cdots)\)

Embedding invariants

Embedding label 99.1
Root \(0.793353 - 0.608761i\) of defining polynomial
Character \(\chi\) \(=\) 776.99
Dual form 776.1.bp.a.243.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.991445 - 0.130526i) q^{2} +(-0.207107 + 0.158919i) q^{3} +(0.965926 + 0.258819i) q^{4} +(0.226078 - 0.130526i) q^{6} +(-0.923880 - 0.382683i) q^{8} +(-0.241181 + 0.900100i) q^{9} +O(q^{10})\) \(q+(-0.991445 - 0.130526i) q^{2} +(-0.207107 + 0.158919i) q^{3} +(0.965926 + 0.258819i) q^{4} +(0.226078 - 0.130526i) q^{6} +(-0.923880 - 0.382683i) q^{8} +(-0.241181 + 0.900100i) q^{9} +(0.0675653 + 0.513210i) q^{11} +(-0.241181 + 0.0999004i) q^{12} +(0.866025 + 0.500000i) q^{16} +(-0.128293 + 1.95737i) q^{17} +(0.356604 - 0.860919i) q^{18} +(0.923880 - 1.38268i) q^{19} -0.517638i q^{22} +(0.252157 - 0.0675653i) q^{24} +(-0.608761 + 0.793353i) q^{25} +(-0.192993 - 0.465926i) q^{27} +(-0.793353 - 0.608761i) q^{32} +(-0.0955518 - 0.0955518i) q^{33} +(0.382683 - 1.92388i) q^{34} +(-0.465926 + 0.807007i) q^{36} +(-1.09645 + 1.25026i) q^{38} +(1.05217 + 0.357164i) q^{41} +(0.410670 + 1.53264i) q^{43} +(-0.0675653 + 0.513210i) q^{44} +(-0.258819 + 0.0340742i) q^{48} +(0.991445 - 0.130526i) q^{49} +(0.707107 - 0.707107i) q^{50} +(-0.284492 - 0.425773i) q^{51} +(0.130526 + 0.487130i) q^{54} +(0.0283924 + 0.433185i) q^{57} +(-0.257264 + 0.293353i) q^{59} +(0.707107 + 0.707107i) q^{64} +(0.0822623 + 0.107206i) q^{66} +(-1.57469 - 1.05217i) q^{67} +(-0.630526 + 1.85747i) q^{68} +(0.567275 - 0.739288i) q^{72} +(1.36603 - 0.366025i) q^{73} -0.261052i q^{75} +(1.25026 - 1.09645i) q^{76} +(-0.692993 - 0.400100i) q^{81} +(-0.996552 - 0.491445i) q^{82} +(1.50046 + 0.0983454i) q^{83} +(-0.207107 - 1.57313i) q^{86} +(0.133975 - 0.500000i) q^{88} +(-1.12484 - 0.465926i) q^{89} +0.261052 q^{96} +(-0.793353 + 0.608761i) q^{97} -1.00000 q^{98} +(-0.478235 - 0.0629609i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{3} - 8 q^{9} - 8 q^{12} - 16 q^{27} + 8 q^{36} + 8 q^{66} - 8 q^{68} + 8 q^{73} - 24 q^{81} + 8 q^{86} + 16 q^{88} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/776\mathbb{Z}\right)^\times\).

\(n\) \(389\) \(393\) \(583\)
\(\chi(n)\) \(-1\) \(e\left(\frac{17}{48}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.991445 0.130526i −0.991445 0.130526i
\(3\) −0.207107 + 0.158919i −0.207107 + 0.158919i −0.707107 0.707107i \(-0.750000\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(5\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(6\) 0.226078 0.130526i 0.226078 0.130526i
\(7\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(8\) −0.923880 0.382683i −0.923880 0.382683i
\(9\) −0.241181 + 0.900100i −0.241181 + 0.900100i
\(10\) 0 0
\(11\) 0.0675653 + 0.513210i 0.0675653 + 0.513210i 0.991445 + 0.130526i \(0.0416667\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(12\) −0.241181 + 0.0999004i −0.241181 + 0.0999004i
\(13\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(17\) −0.128293 + 1.95737i −0.128293 + 1.95737i 0.130526 + 0.991445i \(0.458333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(18\) 0.356604 0.860919i 0.356604 0.860919i
\(19\) 0.923880 1.38268i 0.923880 1.38268i 1.00000i \(-0.5\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.517638i 0.517638i
\(23\) 0 0 −0.659346 0.751840i \(-0.729167\pi\)
0.659346 + 0.751840i \(0.270833\pi\)
\(24\) 0.252157 0.0675653i 0.252157 0.0675653i
\(25\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(26\) 0 0
\(27\) −0.192993 0.465926i −0.192993 0.465926i
\(28\) 0 0
\(29\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(30\) 0 0
\(31\) 0 0 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(32\) −0.793353 0.608761i −0.793353 0.608761i
\(33\) −0.0955518 0.0955518i −0.0955518 0.0955518i
\(34\) 0.382683 1.92388i 0.382683 1.92388i
\(35\) 0 0
\(36\) −0.465926 + 0.807007i −0.465926 + 0.807007i
\(37\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(38\) −1.09645 + 1.25026i −1.09645 + 1.25026i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.05217 + 0.357164i 1.05217 + 0.357164i 0.793353 0.608761i \(-0.208333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(42\) 0 0
\(43\) 0.410670 + 1.53264i 0.410670 + 1.53264i 0.793353 + 0.608761i \(0.208333\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(44\) −0.0675653 + 0.513210i −0.0675653 + 0.513210i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −0.258819 + 0.0340742i −0.258819 + 0.0340742i
\(49\) 0.991445 0.130526i 0.991445 0.130526i
\(50\) 0.707107 0.707107i 0.707107 0.707107i
\(51\) −0.284492 0.425773i −0.284492 0.425773i
\(52\) 0 0
\(53\) 0 0 0.130526 0.991445i \(-0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(54\) 0.130526 + 0.487130i 0.130526 + 0.487130i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.0283924 + 0.433185i 0.0283924 + 0.433185i
\(58\) 0 0
\(59\) −0.257264 + 0.293353i −0.257264 + 0.293353i −0.866025 0.500000i \(-0.833333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(65\) 0 0
\(66\) 0.0822623 + 0.107206i 0.0822623 + 0.107206i
\(67\) −1.57469 1.05217i −1.57469 1.05217i −0.965926 0.258819i \(-0.916667\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(68\) −0.630526 + 1.85747i −0.630526 + 1.85747i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(72\) 0.567275 0.739288i 0.567275 0.739288i
\(73\) 1.36603 0.366025i 1.36603 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) 0 0
\(75\) 0.261052i 0.261052i
\(76\) 1.25026 1.09645i 1.25026 1.09645i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(80\) 0 0
\(81\) −0.692993 0.400100i −0.692993 0.400100i
\(82\) −0.996552 0.491445i −0.996552 0.491445i
\(83\) 1.50046 + 0.0983454i 1.50046 + 0.0983454i 0.793353 0.608761i \(-0.208333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.207107 1.57313i −0.207107 1.57313i
\(87\) 0 0
\(88\) 0.133975 0.500000i 0.133975 0.500000i
\(89\) −1.12484 0.465926i −1.12484 0.465926i −0.258819 0.965926i \(-0.583333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.261052 0.261052
\(97\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(98\) −1.00000 −1.00000
\(99\) −0.478235 0.0629609i −0.478235 0.0629609i
\(100\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(101\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(102\) 0.226484 + 0.459264i 0.226484 + 0.459264i
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.641502 1.88981i −0.641502 1.88981i −0.382683 0.923880i \(-0.625000\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(108\) −0.0658262 0.500000i −0.0658262 0.500000i
\(109\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.05441 0.608761i −1.05441 0.608761i −0.130526 0.991445i \(-0.541667\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(114\) 0.0283924 0.433185i 0.0283924 0.433185i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0.293353 0.257264i 0.293353 0.257264i
\(119\) 0 0
\(120\) 0 0
\(121\) 0.707107 0.189469i 0.707107 0.189469i
\(122\) 0 0
\(123\) −0.274672 + 0.0932386i −0.274672 + 0.0932386i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(128\) −0.608761 0.793353i −0.608761 0.793353i
\(129\) −0.328618 0.252157i −0.328618 0.252157i
\(130\) 0 0
\(131\) 0.389345 1.95737i 0.389345 1.95737i 0.130526 0.991445i \(-0.458333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(132\) −0.0675653 0.117027i −0.0675653 0.117027i
\(133\) 0 0
\(134\) 1.42388 + 1.24871i 1.42388 + 1.24871i
\(135\) 0 0
\(136\) 0.867580 1.75928i 0.867580 1.75928i
\(137\) −0.117317 1.78990i −0.117317 1.78990i −0.500000 0.866025i \(-0.666667\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(138\) 0 0
\(139\) −1.49144 + 0.996552i −1.49144 + 0.996552i −0.500000 + 0.866025i \(0.666667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.658919 + 0.658919i −0.658919 + 0.658919i
\(145\) 0 0
\(146\) −1.40211 + 0.184592i −1.40211 + 0.184592i
\(147\) −0.184592 + 0.184592i −0.184592 + 0.184592i
\(148\) 0 0
\(149\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(150\) −0.0340742 + 0.258819i −0.0340742 + 0.258819i
\(151\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(152\) −1.38268 + 0.923880i −1.38268 + 0.923880i
\(153\) −1.73089 0.587557i −1.73089 0.587557i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0.634841 + 0.487130i 0.634841 + 0.487130i
\(163\) −0.608761 0.793353i −0.608761 0.793353i 0.382683 0.923880i \(-0.375000\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(164\) 0.923880 + 0.617317i 0.923880 + 0.617317i
\(165\) 0 0
\(166\) −1.47479 0.293353i −1.47479 0.293353i
\(167\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(168\) 0 0
\(169\) 0.608761 0.793353i 0.608761 0.793353i
\(170\) 0 0
\(171\) 1.02173 + 1.16506i 1.02173 + 1.16506i
\(172\) 1.58671i 1.58671i
\(173\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.198092 + 0.478235i −0.198092 + 0.478235i
\(177\) 0.00666180 0.101640i 0.00666180 0.101640i
\(178\) 1.05441 + 0.608761i 1.05441 + 0.608761i
\(179\) 0.349942 + 0.172572i 0.349942 + 0.172572i 0.608761 0.793353i \(-0.291667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(180\) 0 0
\(181\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.01321 + 0.0664093i −1.01321 + 0.0664093i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(192\) −0.258819 0.0340742i −0.258819 0.0340742i
\(193\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0.866025 0.500000i 0.866025 0.500000i
\(195\) 0 0
\(196\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(197\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(198\) 0.465926 + 0.124844i 0.465926 + 0.124844i
\(199\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(200\) 0.866025 0.500000i 0.866025 0.500000i
\(201\) 0.493338 0.0323351i 0.493338 0.0323351i
\(202\) 0 0
\(203\) 0 0
\(204\) −0.164600 0.484897i −0.164600 0.484897i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.772029 + 0.380722i 0.772029 + 0.380722i
\(210\) 0 0
\(211\) 0.0420463 0.641502i 0.0420463 0.641502i −0.923880 0.382683i \(-0.875000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0.389345 + 1.95737i 0.389345 + 1.95737i
\(215\) 0 0
\(216\) 0.504314i 0.504314i
\(217\) 0 0
\(218\) 0 0
\(219\) −0.224745 + 0.292893i −0.224745 + 0.292893i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(224\) 0 0
\(225\) −0.567275 0.739288i −0.567275 0.739288i
\(226\) 0.965926 + 0.741181i 0.965926 + 0.741181i
\(227\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(228\) −0.0846915 + 0.425773i −0.0846915 + 0.425773i
\(229\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.583242 1.18270i 0.583242 1.18270i −0.382683 0.923880i \(-0.625000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.324423 + 0.216773i −0.324423 + 0.216773i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(240\) 0 0
\(241\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(242\) −0.725788 + 0.0955518i −0.725788 + 0.0955518i
\(243\) 0.707107 0.0930924i 0.707107 0.0930924i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.284492 0.0565890i 0.284492 0.0565890i
\(247\) 0 0
\(248\) 0 0
\(249\) −0.326384 + 0.218083i −0.326384 + 0.218083i
\(250\) 0 0
\(251\) 0.0726721 + 1.10876i 0.0726721 + 1.10876i 0.866025 + 0.500000i \(0.166667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(257\) 0.0255190 0.128293i 0.0255190 0.128293i −0.965926 0.258819i \(-0.916667\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(258\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −0.641502 + 1.88981i −0.641502 + 1.88981i
\(263\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(264\) 0.0517123 + 0.124844i 0.0517123 + 0.124844i
\(265\) 0 0
\(266\) 0 0
\(267\) 0.307007 0.0822623i 0.307007 0.0822623i
\(268\) −1.24871 1.42388i −1.24871 1.42388i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(272\) −1.08979 + 1.63099i −1.08979 + 1.63099i
\(273\) 0 0
\(274\) −0.117317 + 1.78990i −0.117317 + 1.78990i
\(275\) −0.448288 0.258819i −0.448288 0.258819i
\(276\) 0 0
\(277\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(278\) 1.60876 0.793353i 1.60876 0.793353i
\(279\) 0 0
\(280\) 0 0
\(281\) 0.608761 + 1.79335i 0.608761 + 1.79335i 0.608761 + 0.793353i \(0.291667\pi\)
1.00000i \(0.500000\pi\)
\(282\) 0 0
\(283\) 1.30656 + 0.541196i 1.30656 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.739288 0.567275i 0.739288 0.567275i
\(289\) −2.82340 0.371707i −2.82340 0.371707i
\(290\) 0 0
\(291\) 0.0675653 0.252157i 0.0675653 0.252157i
\(292\) 1.41421 1.41421
\(293\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(294\) 0.207107 0.158919i 0.207107 0.158919i
\(295\) 0 0
\(296\) 0 0
\(297\) 0.226078 0.130526i 0.226078 0.130526i
\(298\) 0 0
\(299\) 0 0
\(300\) 0.0675653 0.252157i 0.0675653 0.252157i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.49144 0.735499i 1.49144 0.735499i
\(305\) 0 0
\(306\) 1.63939 + 0.808456i 1.63939 + 0.808456i
\(307\) −1.71723 0.991445i −1.71723 0.991445i −0.923880 0.382683i \(-0.875000\pi\)
−0.793353 0.608761i \(-0.791667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(312\) 0 0
\(313\) 1.58671i 1.58671i 0.608761 + 0.793353i \(0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.433185 + 0.289445i 0.433185 + 0.289445i
\(322\) 0 0
\(323\) 2.58790 + 1.98576i 2.58790 + 1.98576i
\(324\) −0.565826 0.565826i −0.565826 0.565826i
\(325\) 0 0
\(326\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(327\) 0 0
\(328\) −0.835400 0.732626i −0.835400 0.732626i
\(329\) 0 0
\(330\) 0 0
\(331\) 0.0578541 + 0.882683i 0.0578541 + 0.882683i 0.923880 + 0.382683i \(0.125000\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(332\) 1.42388 + 0.483342i 1.42388 + 0.483342i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.732626 1.09645i −0.732626 1.09645i −0.991445 0.130526i \(-0.958333\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(338\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(339\) 0.315118 0.0414861i 0.315118 0.0414861i
\(340\) 0 0
\(341\) 0 0
\(342\) −0.860919 1.28846i −0.860919 1.28846i
\(343\) 0 0
\(344\) 0.207107 1.57313i 0.207107 1.57313i
\(345\) 0 0
\(346\) 0 0
\(347\) 1.24871 + 0.423880i 1.24871 + 0.423880i 0.866025 0.500000i \(-0.166667\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(348\) 0 0
\(349\) 0 0 0.442289 0.896873i \(-0.354167\pi\)
−0.442289 + 0.896873i \(0.645833\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.258819 0.448288i 0.258819 0.448288i
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) −0.0198714 + 0.0999004i −0.0198714 + 0.0999004i
\(355\) 0 0
\(356\) −0.965926 0.741181i −0.965926 0.741181i
\(357\) 0 0
\(358\) −0.324423 0.216773i −0.324423 0.216773i
\(359\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(360\) 0 0
\(361\) −0.675577 1.63099i −0.675577 1.63099i
\(362\) 0 0
\(363\) −0.116337 + 0.151613i −0.116337 + 0.151613i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(368\) 0 0
\(369\) −0.575247 + 0.860919i −0.575247 + 0.860919i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.896873 0.442289i \(-0.854167\pi\)
0.896873 + 0.442289i \(0.145833\pi\)
\(374\) 1.01321 + 0.0664093i 1.01321 + 0.0664093i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.448288 + 1.67303i −0.448288 + 1.67303i 0.258819 + 0.965926i \(0.416667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(384\) 0.252157 + 0.0675653i 0.252157 + 0.0675653i
\(385\) 0 0
\(386\) −0.991445 0.130526i −0.991445 0.130526i
\(387\) −1.47858 −1.47858
\(388\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.965926 0.258819i −0.965926 0.258819i
\(393\) 0.230427 + 0.467259i 0.230427 + 0.467259i
\(394\) 0 0
\(395\) 0 0
\(396\) −0.445644 0.184592i −0.445644 0.184592i
\(397\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(401\) 0.793353 0.391239i 0.793353 0.391239i 1.00000i \(-0.5\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(402\) −0.493338 0.0323351i −0.493338 0.0323351i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0.0999004 + 0.502233i 0.0999004 + 0.502233i
\(409\) 1.13053 0.991445i 1.13053 0.991445i 0.130526 0.991445i \(-0.458333\pi\)
1.00000 \(0\)
\(410\) 0 0
\(411\) 0.308746 + 0.352058i 0.308746 + 0.352058i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.150518 0.443411i 0.150518 0.443411i
\(418\) −0.715730 0.478235i −0.715730 0.478235i
\(419\) −1.20711 1.57313i −1.20711 1.57313i −0.707107 0.707107i \(-0.750000\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(420\) 0 0
\(421\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(422\) −0.125419 + 0.630526i −0.125419 + 0.630526i
\(423\) 0 0
\(424\) 0 0
\(425\) −1.47479 1.29335i −1.47479 1.29335i
\(426\) 0 0
\(427\) 0 0
\(428\) −0.130526 1.99144i −0.130526 1.99144i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(432\) 0.0658262 0.500000i 0.0658262 0.500000i
\(433\) 1.85747 0.369474i 1.85747 0.369474i 0.866025 0.500000i \(-0.166667\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0.261052 0.261052i 0.261052 0.261052i
\(439\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(440\) 0 0
\(441\) −0.121631 + 0.923880i −0.121631 + 0.923880i
\(442\) 0 0
\(443\) 0.735499 0.491445i 0.735499 0.491445i −0.130526 0.991445i \(-0.541667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.991445 1.71723i 0.991445 1.71723i 0.382683 0.923880i \(-0.375000\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(450\) 0.465926 + 0.807007i 0.465926 + 0.807007i
\(451\) −0.112210 + 0.564117i −0.112210 + 0.564117i
\(452\) −0.860919 0.860919i −0.860919 0.860919i
\(453\) 0 0
\(454\) −1.05441 1.37413i −1.05441 1.37413i
\(455\) 0 0
\(456\) 0.139541 0.411076i 0.139541 0.411076i
\(457\) 0.128293 + 0.0255190i 0.128293 + 0.0255190i 0.258819 0.965926i \(-0.416667\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(458\) 0 0
\(459\) 0.936749 0.317983i 0.936749 0.317983i
\(460\) 0 0
\(461\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.732626 + 1.09645i −0.732626 + 1.09645i
\(467\) −0.292893 + 0.707107i −0.292893 + 0.707107i 0.707107 + 0.707107i \(0.250000\pi\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0.349942 0.172572i 0.349942 0.172572i
\(473\) −0.758819 + 0.314313i −0.758819 + 0.314313i
\(474\) 0 0
\(475\) 0.534534 + 1.57469i 0.534534 + 1.57469i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0.732051 0.732051
\(485\) 0 0
\(486\) −0.713208 −0.713208
\(487\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(488\) 0 0
\(489\) 0.252157 + 0.0675653i 0.252157 + 0.0675653i
\(490\) 0 0
\(491\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(492\) −0.289445 + 0.0189712i −0.289445 + 0.0189712i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0.352058 0.173616i 0.352058 0.173616i
\(499\) −1.65938 0.108761i −1.65938 0.108761i −0.793353 0.608761i \(-0.791667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.0726721 1.10876i 0.0726721 1.10876i
\(503\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.261052i 0.261052i
\(508\) 0 0
\(509\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.382683 0.923880i −0.382683 0.923880i
\(513\) −0.822530 0.163611i −0.822530 0.163611i
\(514\) −0.0420463 + 0.123864i −0.0420463 + 0.123864i
\(515\) 0 0
\(516\) −0.252157 0.328618i −0.252157 0.328618i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −0.869474 + 0.991445i −0.869474 + 0.991445i 0.130526 + 0.991445i \(0.458333\pi\)
−1.00000 \(\pi\)
\(524\) 0.882683 1.78990i 0.882683 1.78990i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −0.0349744 0.130526i −0.0349744 0.130526i
\(529\) −0.130526 + 0.991445i −0.130526 + 0.991445i
\(530\) 0 0
\(531\) −0.202000 0.302314i −0.202000 0.302314i
\(532\) 0 0
\(533\) 0 0
\(534\) −0.315118 + 0.0414861i −0.315118 + 0.0414861i
\(535\) 0 0
\(536\) 1.05217 + 1.57469i 1.05217 + 1.57469i
\(537\) −0.0999004 + 0.0198714i −0.0999004 + 0.0198714i
\(538\) 0 0
\(539\) 0.133975 + 0.500000i 0.133975 + 0.500000i
\(540\) 0 0
\(541\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.29335 1.47479i 1.29335 1.47479i
\(545\) 0 0
\(546\) 0 0
\(547\) 0.793353 + 1.37413i 0.793353 + 1.37413i 0.923880 + 0.382683i \(0.125000\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(548\) 0.349942 1.75928i 0.349942 1.75928i
\(549\) 0 0
\(550\) 0.410670 + 0.315118i 0.410670 + 0.315118i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.69855 + 0.576581i −1.69855 + 0.576581i
\(557\) 0 0 0.608761 0.793353i \(-0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0.199289 0.174772i 0.199289 0.174772i
\(562\) −0.369474 1.85747i −0.369474 1.85747i
\(563\) 1.08979 1.63099i 1.08979 1.63099i 0.382683 0.923880i \(-0.375000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.22474 0.707107i −1.22474 0.707107i
\(567\) 0 0
\(568\) 0 0
\(569\) −0.576581 + 0.284338i −0.576581 + 0.284338i −0.707107 0.707107i \(-0.750000\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(570\) 0 0
\(571\) −0.258819 1.96593i −0.258819 1.96593i −0.258819 0.965926i \(-0.583333\pi\)
1.00000i \(-0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.807007 + 0.465926i −0.807007 + 0.465926i
\(577\) 0.284338 + 0.576581i 0.284338 + 0.576581i 0.991445 0.130526i \(-0.0416667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(578\) 2.75072 + 0.737054i 2.75072 + 0.737054i
\(579\) −0.207107 + 0.158919i −0.207107 + 0.158919i
\(580\) 0 0
\(581\) 0 0
\(582\) −0.0999004 + 0.241181i −0.0999004 + 0.241181i
\(583\) 0 0
\(584\) −1.40211 0.184592i −1.40211 0.184592i
\(585\) 0 0
\(586\) 0 0
\(587\) 0.391239 + 0.793353i 0.391239 + 0.793353i 1.00000 \(0\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(588\) −0.226078 + 0.130526i −0.226078 + 0.130526i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.130526 + 0.991445i 0.130526 + 0.991445i 0.923880 + 0.382683i \(0.125000\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(594\) −0.241181 + 0.0999004i −0.241181 + 0.0999004i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.0654031 0.997859i \(-0.479167\pi\)
−0.0654031 + 0.997859i \(0.520833\pi\)
\(600\) −0.0999004 + 0.241181i −0.0999004 + 0.241181i
\(601\) 0.216773 0.324423i 0.216773 0.324423i −0.707107 0.707107i \(-0.750000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(602\) 0 0
\(603\) 1.32684 1.16361i 1.32684 1.16361i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.608761 0.793353i \(-0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(608\) −1.57469 + 0.534534i −1.57469 + 0.534534i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −1.51984 1.01552i −1.51984 1.01552i
\(613\) 0 0 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(614\) 1.57313 + 1.20711i 1.57313 + 1.20711i
\(615\) 0 0
\(616\) 0 0
\(617\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(618\) 0 0
\(619\) 0.835400 + 0.732626i 0.835400 + 0.732626i 0.965926 0.258819i \(-0.0833333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.258819 0.965926i −0.258819 0.965926i
\(626\) 0.207107 1.57313i 0.207107 1.57313i
\(627\) −0.220396 + 0.0438395i −0.220396 + 0.0438395i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(632\) 0 0
\(633\) 0.0932386 + 0.139541i 0.0932386 + 0.139541i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.18270 1.34861i 1.18270 1.34861i 0.258819 0.965926i \(-0.416667\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(642\) −0.391699 0.343511i −0.391699 0.343511i
\(643\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2.30656 2.30656i −2.30656 2.30656i
\(647\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(648\) 0.487130 + 0.634841i 0.487130 + 0.634841i
\(649\) −0.167934 0.112210i −0.167934 0.112210i
\(650\) 0 0
\(651\) 0 0
\(652\) −0.382683 0.923880i −0.382683 0.923880i
\(653\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.732626 + 0.835400i 0.732626 + 0.835400i
\(657\) 1.31784i 1.31784i
\(658\) 0 0
\(659\) 0.0255190 + 0.128293i 0.0255190 + 0.128293i 0.991445 0.130526i \(-0.0416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(660\) 0 0
\(661\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(662\) 0.0578541 0.882683i 0.0578541 0.882683i
\(663\) 0 0
\(664\) −1.34861 0.665060i −1.34861 0.665060i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.71723 + 0.991445i −1.71723 + 0.991445i −0.793353 + 0.608761i \(0.791667\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(674\) 0.583242 + 1.18270i 0.583242 + 1.18270i
\(675\) 0.487130 + 0.130526i 0.487130 + 0.130526i
\(676\) 0.793353 0.608761i 0.793353 0.608761i
\(677\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(678\) −0.317837 −0.317837
\(679\) 0 0
\(680\) 0 0
\(681\) −0.448288 0.0590182i −0.448288 0.0590182i
\(682\) 0 0
\(683\) 1.91532 + 0.513210i 1.91532 + 0.513210i 0.991445 + 0.130526i \(0.0416667\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(684\) 0.685376 + 1.38981i 0.685376 + 1.38981i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −0.410670 + 1.53264i −0.410670 + 1.53264i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.707107 + 0.292893i −0.707107 + 0.292893i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.18270 0.583242i −1.18270 0.583242i
\(695\) 0 0
\(696\) 0 0
\(697\) −0.834089 + 2.01367i −0.834089 + 2.01367i
\(698\) 0 0
\(699\) 0.0671594 + 0.337633i 0.0671594 + 0.337633i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.315118 + 0.410670i −0.315118 + 0.410670i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0.0327410 0.0964520i 0.0327410 0.0964520i
\(709\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.860919 + 0.860919i 0.860919 + 0.860919i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.293353 + 0.257264i 0.293353 + 0.257264i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.0654031 0.997859i \(-0.520833\pi\)
0.0654031 + 0.997859i \(0.479167\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.456911 + 1.70521i 0.456911 + 1.70521i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0.135131 0.135131i 0.135131 0.135131i
\(727\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(728\) 0 0
\(729\) 0.434174 0.434174i 0.434174 0.434174i
\(730\) 0 0
\(731\) −3.05263 + 0.607206i −3.05263 + 0.607206i
\(732\) 0 0
\(733\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.433591 0.879235i 0.433591 0.879235i
\(738\) 0.682699 0.778469i 0.682699 0.778469i
\(739\) −1.34861 1.18270i −1.34861 1.18270i −0.965926 0.258819i \(-0.916667\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.450403 + 1.32684i −0.450403 + 1.32684i
\(748\) −0.995873 0.198092i −0.995873 0.198092i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.608761 0.793353i \(-0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(752\) 0 0
\(753\) −0.191254 0.218083i −0.191254 0.218083i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(758\) 0.662827 1.60021i 0.662827 1.60021i
\(759\) 0 0
\(760\) 0 0
\(761\) 1.75928 + 0.867580i 1.75928 + 0.867580i 0.965926 + 0.258819i \(0.0833333\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.241181 0.0999004i −0.241181 0.0999004i
\(769\) 1.65938 0.108761i 1.65938 0.108761i 0.793353 0.608761i \(-0.208333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(770\) 0 0
\(771\) 0.0151030 + 0.0306258i 0.0151030 + 0.0306258i
\(772\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(773\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(774\) 1.46593 + 0.192993i 1.46593 + 0.192993i
\(775\) 0 0
\(776\) 0.965926 0.258819i 0.965926 0.258819i
\(777\) 0 0
\(778\) 0 0
\(779\) 1.46593 1.12484i 1.46593 1.12484i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(785\) 0 0
\(786\) −0.167466 0.493338i −0.167466 0.493338i
\(787\) 0.241181 + 1.83195i 0.241181 + 1.83195i 0.500000 + 0.866025i \(0.333333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.417738 + 0.241181i 0.417738 + 0.241181i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.965926 0.258819i 0.965926 0.258819i
\(801\) 0.690671 0.900100i 0.690671 0.900100i
\(802\) −0.837633 + 0.284338i −0.837633 + 0.284338i
\(803\) 0.280144 + 0.676327i 0.280144 + 0.676327i
\(804\) 0.484897 + 0.0964520i 0.484897 + 0.0964520i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(810\) 0 0
\(811\) −0.608761 1.05441i −0.608761 1.05441i −0.991445 0.130526i \(-0.958333\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −0.0334912 0.510976i −0.0334912 0.510976i
\(817\) 2.49857 + 0.848149i 2.49857 + 0.848149i
\(818\) −1.25026 + 0.835400i −1.25026 + 0.835400i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(822\) −0.260152 0.389345i −0.260152 0.389345i
\(823\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(824\) 0 0
\(825\) 0.133975 0.0176381i 0.133975 0.0176381i
\(826\) 0 0
\(827\) 0.357164 + 0.534534i 0.357164 + 0.534534i 0.965926 0.258819i \(-0.0833333\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(828\) 0 0
\(829\) 0 0 0.130526 0.991445i \(-0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.128293 + 1.95737i 0.128293 + 1.95737i
\(834\) −0.207107 + 0.419971i −0.207107 + 0.419971i
\(835\) 0 0
\(836\) 0.647184 + 0.567565i 0.647184 + 0.567565i
\(837\) 0 0
\(838\) 0.991445 + 1.71723i 0.991445 + 1.71723i
\(839\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(840\) 0 0
\(841\) −0.793353 0.608761i −0.793353 0.608761i
\(842\) 0 0
\(843\) −0.411076 0.274672i −0.411076 0.274672i
\(844\) 0.206647 0.608761i 0.206647 0.608761i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.356604 + 0.0955518i −0.356604 + 0.0955518i
\(850\) 1.29335 + 1.47479i 1.29335 + 1.47479i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.130526 + 1.99144i −0.130526 + 1.99144i
\(857\) 0.448288 + 0.258819i 0.448288 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(858\) 0 0
\(859\) −1.99144 0.130526i −1.99144 0.130526i −0.991445 0.130526i \(-0.958333\pi\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.321439 0.946930i \(-0.604167\pi\)
0.321439 + 0.946930i \(0.395833\pi\)
\(864\) −0.130526 + 0.487130i −0.130526 + 0.487130i
\(865\) 0 0
\(866\) −1.88981 + 0.123864i −1.88981 + 0.123864i
\(867\) 0.643816 0.371707i 0.643816 0.371707i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.356604 0.860919i −0.356604 0.860919i
\(874\) 0 0
\(875\) 0 0
\(876\) −0.292893 + 0.224745i −0.292893 + 0.224745i
\(877\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.60021 0.662827i −1.60021 0.662827i −0.608761 0.793353i \(-0.708333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(882\) 0.241181 0.900100i 0.241181 0.900100i
\(883\) 0.641502 + 1.88981i 0.641502 + 1.88981i 0.382683 + 0.923880i \(0.375000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −0.793353 + 0.391239i −0.793353 + 0.391239i
\(887\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.158513 0.382683i 0.158513 0.382683i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.20711 + 1.57313i −1.20711 + 1.57313i
\(899\) 0 0
\(900\) −0.356604 0.860919i −0.356604 0.860919i
\(901\) 0 0
\(902\) 0.184882 0.544645i 0.184882 0.544645i
\(903\) 0 0
\(904\) 0.741181 + 0.965926i 0.741181 + 0.965926i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.293353 1.47479i 0.293353 1.47479i −0.500000 0.866025i \(-0.666667\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(908\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(912\) −0.192004 + 0.389345i −0.192004 + 0.389345i
\(913\) 0.0509073 + 0.776695i 0.0509073 + 0.776695i
\(914\) −0.123864 0.0420463i −0.123864 0.0420463i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −0.970240 + 0.192993i −0.970240 + 0.192993i
\(919\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(920\) 0 0
\(921\) 0.513210 0.0675653i 0.513210 0.0675653i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −0.123864 0.0420463i −0.123864 0.0420463i 0.258819 0.965926i \(-0.416667\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(930\) 0 0
\(931\) 0.735499 1.49144i 0.735499 1.49144i
\(932\) 0.869474 0.991445i 0.869474 0.991445i
\(933\) 0 0
\(934\) 0.382683 0.662827i 0.382683 0.662827i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(938\) 0 0
\(939\) −0.252157 0.328618i −0.252157 0.328618i
\(940\) 0 0
\(941\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −0.369474 + 0.125419i −0.369474 + 0.125419i
\(945\) 0 0
\(946\) 0.793353 0.212578i 0.793353 0.212578i
\(947\) −0.991445 1.13053i −0.991445 1.13053i −0.991445 0.130526i \(-0.958333\pi\)
1.00000i \(-0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −0.324423 1.63099i −0.324423 1.63099i
\(951\) 0 0
\(952\) 0 0
\(953\) −0.0255190 + 0.389345i −0.0255190 + 0.389345i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(962\) 0 0
\(963\) 1.85573 0.121631i 1.85573 0.121631i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(968\) −0.725788 0.0955518i −0.725788 0.0955518i
\(969\) −0.851546 −0.851546
\(970\) 0 0
\(971\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(972\) 0.707107 + 0.0930924i 0.707107 + 0.0930924i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.95737 + 0.128293i −1.95737 + 0.128293i −0.991445 0.130526i \(-0.958333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(978\) −0.241181 0.0999004i −0.241181 0.0999004i
\(979\) 0.163117 0.608761i 0.163117 0.608761i
\(980\) 0 0
\(981\) 0 0
\(982\) 0.923880 0.382683i 0.923880 0.382683i
\(983\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(984\) 0.289445 + 0.0189712i 0.289445 + 0.0189712i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(992\) 0 0
\(993\) −0.152257 0.173616i −0.152257 0.173616i
\(994\) 0 0
\(995\) 0 0
\(996\) −0.371707 + 0.126178i −0.371707 + 0.126178i
\(997\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(998\) 1.63099 + 0.324423i 1.63099 + 0.324423i
\(999\) 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 776.1.bp.a.99.1 16
4.3 odd 2 3104.1.et.a.1263.1 16
8.3 odd 2 CM 776.1.bp.a.99.1 16
8.5 even 2 3104.1.et.a.1263.1 16
97.49 even 48 inner 776.1.bp.a.243.1 yes 16
388.243 odd 48 3104.1.et.a.2959.1 16
776.243 odd 48 inner 776.1.bp.a.243.1 yes 16
776.437 even 48 3104.1.et.a.2959.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
776.1.bp.a.99.1 16 1.1 even 1 trivial
776.1.bp.a.99.1 16 8.3 odd 2 CM
776.1.bp.a.243.1 yes 16 97.49 even 48 inner
776.1.bp.a.243.1 yes 16 776.243 odd 48 inner
3104.1.et.a.1263.1 16 4.3 odd 2
3104.1.et.a.1263.1 16 8.5 even 2
3104.1.et.a.2959.1 16 388.243 odd 48
3104.1.et.a.2959.1 16 776.437 even 48