Properties

Label 884.1.ct.a.687.1
Level $884$
Weight $1$
Character 884.687
Analytic conductor $0.441$
Analytic rank $0$
Dimension $16$
Projective image $D_{48}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [884,1,Mod(7,884)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(884, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([24, 44, 33]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("884.7");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 884 = 2^{2} \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 884.ct (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: \(0.441173471168\)
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 687.1
Root \(0.130526 - 0.991445i\) of defining polynomial
Character \(\chi\) \(=\) 884.687
Dual form 884.1.ct.a.175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.793353 + 0.608761i) q^{2} +(0.258819 + 0.965926i) q^{4} +(-0.357164 + 0.534534i) q^{5} +(-0.382683 + 0.923880i) q^{8} +(0.991445 + 0.130526i) q^{9} +O(q^{10})\) \(q+(0.793353 + 0.608761i) q^{2} +(0.258819 + 0.965926i) q^{4} +(-0.357164 + 0.534534i) q^{5} +(-0.382683 + 0.923880i) q^{8} +(0.991445 + 0.130526i) q^{9} +(-0.608761 + 0.206647i) q^{10} +(-0.130526 - 0.991445i) q^{13} +(-0.866025 + 0.500000i) q^{16} +(-0.707107 + 0.707107i) q^{17} +(0.707107 + 0.707107i) q^{18} +(-0.608761 - 0.206647i) q^{20} +(0.224523 + 0.542046i) q^{25} +(0.500000 - 0.866025i) q^{26} +(-0.869474 - 0.991445i) q^{29} +(-0.991445 - 0.130526i) q^{32} +(-0.991445 + 0.130526i) q^{34} +(0.130526 + 0.991445i) q^{36} +(0.665060 - 0.583242i) q^{37} +(-0.357164 - 0.534534i) q^{40} +(-0.117317 - 0.0578541i) q^{41} +(-0.423880 + 0.483342i) q^{45} +(0.991445 - 0.130526i) q^{49} +(-0.151851 + 0.566715i) q^{50} +(0.923880 - 0.382683i) q^{52} +(0.465926 - 1.12484i) q^{53} +(-0.0862466 - 1.31587i) q^{58} +(1.18270 - 1.34861i) q^{61} +(-0.707107 - 0.707107i) q^{64} +(0.576581 + 0.284338i) q^{65} +(-0.866025 - 0.500000i) q^{68} +(-0.500000 + 0.866025i) q^{72} +(-1.65938 - 1.10876i) q^{73} +(0.882683 - 0.0578541i) q^{74} +(0.0420463 - 0.641502i) q^{80} +(0.965926 + 0.258819i) q^{81} +(-0.0578541 - 0.117317i) q^{82} +(-0.125419 - 0.630526i) q^{85} +(-0.382683 - 0.662827i) q^{89} +(-0.630526 + 0.125419i) q^{90} +(-1.75928 + 0.867580i) q^{97} +(0.866025 + 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{26} - 16 q^{29} - 8 q^{41} + 8 q^{45} - 8 q^{53} - 8 q^{72} + 8 q^{74} - 8 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/884\mathbb{Z}\right)^\times\).

\(n\) \(105\) \(443\) \(613\)
\(\chi(n)\) \(e\left(\frac{11}{16}\right)\) \(-1\) \(e\left(\frac{7}{12}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(3\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(4\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(5\) −0.357164 + 0.534534i −0.357164 + 0.534534i −0.965926 0.258819i \(-0.916667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(6\) 0 0
\(7\) 0 0 0.997859 0.0654031i \(-0.0208333\pi\)
−0.997859 + 0.0654031i \(0.979167\pi\)
\(8\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(9\) 0.991445 + 0.130526i 0.991445 + 0.130526i
\(10\) −0.608761 + 0.206647i −0.608761 + 0.206647i
\(11\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(12\) 0 0
\(13\) −0.130526 0.991445i −0.130526 0.991445i
\(14\) 0 0
\(15\) 0 0
\(16\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(17\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(18\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(19\) 0 0 0.130526 0.991445i \(-0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(20\) −0.608761 0.206647i −0.608761 0.206647i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.896873 0.442289i \(-0.854167\pi\)
0.896873 + 0.442289i \(0.145833\pi\)
\(24\) 0 0
\(25\) 0.224523 + 0.542046i 0.224523 + 0.542046i
\(26\) 0.500000 0.866025i 0.500000 0.866025i
\(27\) 0 0
\(28\) 0 0
\(29\) −0.869474 0.991445i −0.869474 0.991445i 0.130526 0.991445i \(-0.458333\pi\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 −0.195090 0.980785i \(-0.562500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(32\) −0.991445 0.130526i −0.991445 0.130526i
\(33\) 0 0
\(34\) −0.991445 + 0.130526i −0.991445 + 0.130526i
\(35\) 0 0
\(36\) 0.130526 + 0.991445i 0.130526 + 0.991445i
\(37\) 0.665060 0.583242i 0.665060 0.583242i −0.258819 0.965926i \(-0.583333\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.357164 0.534534i −0.357164 0.534534i
\(41\) −0.117317 0.0578541i −0.117317 0.0578541i 0.382683 0.923880i \(-0.375000\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0 0 0.130526 0.991445i \(-0.458333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(44\) 0 0
\(45\) −0.423880 + 0.483342i −0.423880 + 0.483342i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0.991445 0.130526i 0.991445 0.130526i
\(50\) −0.151851 + 0.566715i −0.151851 + 0.566715i
\(51\) 0 0
\(52\) 0.923880 0.382683i 0.923880 0.382683i
\(53\) 0.465926 1.12484i 0.465926 1.12484i −0.500000 0.866025i \(-0.666667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.0862466 1.31587i −0.0862466 1.31587i
\(59\) 0 0 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(60\) 0 0
\(61\) 1.18270 1.34861i 1.18270 1.34861i 0.258819 0.965926i \(-0.416667\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.707107 0.707107i −0.707107 0.707107i
\(65\) 0.576581 + 0.284338i 0.576581 + 0.284338i
\(66\) 0 0
\(67\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(68\) −0.866025 0.500000i −0.866025 0.500000i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(72\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(73\) −1.65938 1.10876i −1.65938 1.10876i −0.866025 0.500000i \(-0.833333\pi\)
−0.793353 0.608761i \(-0.791667\pi\)
\(74\) 0.882683 0.0578541i 0.882683 0.0578541i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.831470 0.555570i \(-0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(80\) 0.0420463 0.641502i 0.0420463 0.641502i
\(81\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(82\) −0.0578541 0.117317i −0.0578541 0.117317i
\(83\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(84\) 0 0
\(85\) −0.125419 0.630526i −0.125419 0.630526i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.382683 0.662827i −0.382683 0.662827i 0.608761 0.793353i \(-0.291667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(90\) −0.630526 + 0.125419i −0.630526 + 0.125419i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.75928 + 0.867580i −1.75928 + 0.867580i −0.793353 + 0.608761i \(0.791667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(98\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(99\) 0 0
\(100\) −0.465466 + 0.357164i −0.465466 + 0.357164i
\(101\) −0.130526 0.226078i −0.130526 0.226078i 0.793353 0.608761i \(-0.208333\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(105\) 0 0
\(106\) 1.05441 0.608761i 1.05441 0.608761i
\(107\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(108\) 0 0
\(109\) −0.324423 + 0.216773i −0.324423 + 0.216773i −0.707107 0.707107i \(-0.750000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.0983454 + 1.50046i 0.0983454 + 1.50046i 0.707107 + 0.707107i \(0.250000\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.732626 1.09645i 0.732626 1.09645i
\(117\) 1.00000i 1.00000i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.793353 0.608761i 0.793353 0.608761i
\(122\) 1.75928 0.349942i 1.75928 0.349942i
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00046 0.199004i −1.00046 0.199004i
\(126\) 0 0
\(127\) 0 0 0.608761 0.793353i \(-0.291667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(128\) −0.130526 0.991445i −0.130526 0.991445i
\(129\) 0 0
\(130\) 0.284338 + 0.576581i 0.284338 + 0.576581i
\(131\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.382683 0.923880i −0.382683 0.923880i
\(137\) 0.513210 + 1.91532i 0.513210 + 1.91532i 0.382683 + 0.923880i \(0.375000\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(138\) 0 0
\(139\) 0 0 −0.442289 0.896873i \(-0.645833\pi\)
0.442289 + 0.896873i \(0.354167\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(145\) 0.840506 0.110655i 0.840506 0.110655i
\(146\) −0.641502 1.88981i −0.641502 1.88981i
\(147\) 0 0
\(148\) 0.735499 + 0.491445i 0.735499 + 0.491445i
\(149\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(152\) 0 0
\(153\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.366025 0.366025i −0.366025 0.366025i 0.500000 0.866025i \(-0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.423880 0.483342i 0.423880 0.483342i
\(161\) 0 0
\(162\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(163\) 0 0 −0.0654031 0.997859i \(-0.520833\pi\)
0.0654031 + 0.997859i \(0.479167\pi\)
\(164\) 0.0255190 0.128293i 0.0255190 0.128293i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.659346 0.751840i \(-0.729167\pi\)
0.659346 + 0.751840i \(0.270833\pi\)
\(168\) 0 0
\(169\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(170\) 0.284338 0.576581i 0.284338 0.576581i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.996552 + 0.491445i −0.996552 + 0.491445i −0.866025 0.500000i \(-0.833333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.0999004 0.758819i 0.0999004 0.758819i
\(179\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(180\) −0.576581 0.284338i −0.576581 0.284338i
\(181\) 1.05217 + 1.57469i 1.05217 + 1.57469i 0.793353 + 0.608761i \(0.208333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.0742271 + 0.563811i 0.0742271 + 0.563811i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(192\) 0 0
\(193\) −0.608761 1.79335i −0.608761 1.79335i −0.608761 0.793353i \(-0.708333\pi\)
1.00000i \(-0.5\pi\)
\(194\) −1.92388 0.382683i −1.92388 0.382683i
\(195\) 0 0
\(196\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(197\) 0.491445 0.996552i 0.491445 0.996552i −0.500000 0.866025i \(-0.666667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(198\) 0 0
\(199\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(200\) −0.586707 −0.586707
\(201\) 0 0
\(202\) 0.0340742 0.258819i 0.0340742 0.258819i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.0728263 0.0420463i 0.0728263 0.0420463i
\(206\) 0 0
\(207\) 0 0
\(208\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(212\) 1.20711 + 0.158919i 1.20711 + 0.158919i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.389345 0.0255190i −0.389345 0.0255190i
\(219\) 0 0
\(220\) 0 0
\(221\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(222\) 0 0
\(223\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(224\) 0 0
\(225\) 0.151851 + 0.566715i 0.151851 + 0.566715i
\(226\) −0.835400 + 1.25026i −0.835400 + 1.25026i
\(227\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(228\) 0 0
\(229\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.24871 0.423880i 1.24871 0.423880i
\(233\) 0.324423 + 1.63099i 0.324423 + 1.63099i 0.707107 + 0.707107i \(0.250000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(234\) 0.608761 0.793353i 0.608761 0.793353i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) 1.42388 + 0.483342i 1.42388 + 0.483342i 0.923880 0.382683i \(-0.125000\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 1.00000 1.00000
\(243\) 0 0
\(244\) 1.60876 + 0.793353i 1.60876 + 0.793353i
\(245\) −0.284338 + 0.576581i −0.284338 + 0.576581i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.672572 0.766922i −0.672572 0.766922i
\(251\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.500000 0.866025i 0.500000 0.866025i
\(257\) −0.252157 1.91532i −0.252157 1.91532i −0.382683 0.923880i \(-0.625000\pi\)
0.130526 0.991445i \(-0.458333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.125419 + 0.630526i −0.125419 + 0.630526i
\(261\) −0.732626 1.09645i −0.732626 1.09645i
\(262\) 0 0
\(263\) 0 0 −0.793353 0.608761i \(-0.791667\pi\)
0.793353 + 0.608761i \(0.208333\pi\)
\(264\) 0 0
\(265\) 0.434856 + 0.650808i 0.434856 + 0.650808i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.349942 + 0.172572i −0.349942 + 0.172572i −0.608761 0.793353i \(-0.708333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(270\) 0 0
\(271\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(272\) 0.258819 0.965926i 0.258819 0.965926i
\(273\) 0 0
\(274\) −0.758819 + 1.83195i −0.758819 + 1.83195i
\(275\) 0 0
\(276\) 0 0
\(277\) −0.608761 + 1.79335i −0.608761 + 1.79335i 1.00000i \(0.5\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.241181 0.0999004i −0.241181 0.0999004i 0.258819 0.965926i \(-0.416667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 0 0 0.442289 0.896873i \(-0.354167\pi\)
−0.442289 + 0.896873i \(0.645833\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.965926 0.258819i −0.965926 0.258819i
\(289\) 1.00000i 1.00000i
\(290\) 0.734181 + 0.423880i 0.734181 + 0.423880i
\(291\) 0 0
\(292\) 0.641502 1.88981i 0.641502 1.88981i
\(293\) −0.965926 + 1.67303i −0.965926 + 1.67303i −0.258819 + 0.965926i \(0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.284338 + 0.837633i 0.284338 + 0.837633i
\(297\) 0 0
\(298\) −1.60021 + 0.662827i −1.60021 + 0.662827i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.298460 + 1.11387i 0.298460 + 1.11387i
\(306\) −1.00000 −1.00000
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(312\) 0 0
\(313\) −0.382683 + 1.92388i −0.382683 + 1.92388i 1.00000i \(0.5\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(314\) −0.0675653 0.513210i −0.0675653 0.513210i
\(315\) 0 0
\(316\) 0 0
\(317\) −1.47479 0.293353i −1.47479 0.293353i −0.608761 0.793353i \(-0.708333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.630526 0.125419i 0.630526 0.125419i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000i 1.00000i
\(325\) 0.508103 0.293353i 0.508103 0.293353i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.0983454 0.0862466i 0.0983454 0.0862466i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(332\) 0 0
\(333\) 0.735499 0.491445i 0.735499 0.491445i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.10876 1.65938i 1.10876 1.65938i 0.500000 0.866025i \(-0.333333\pi\)
0.608761 0.793353i \(-0.291667\pi\)
\(338\) −0.923880 0.382683i −0.923880 0.382683i
\(339\) 0 0
\(340\) 0.576581 0.284338i 0.576581 0.284338i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.08979 0.216773i −1.08979 0.216773i
\(347\) 0 0 −0.751840 0.659346i \(-0.770833\pi\)
0.751840 + 0.659346i \(0.229167\pi\)
\(348\) 0 0
\(349\) 0.241181 + 1.83195i 0.241181 + 1.83195i 0.500000 + 0.866025i \(0.333333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.258819 0.448288i −0.258819 0.448288i 0.707107 0.707107i \(-0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.541196 0.541196i 0.541196 0.541196i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(360\) −0.284338 0.576581i −0.284338 0.576581i
\(361\) −0.965926 0.258819i −0.965926 0.258819i
\(362\) −0.123864 + 1.88981i −0.123864 + 1.88981i
\(363\) 0 0
\(364\) 0 0
\(365\) 1.18534 0.490985i 1.18534 0.490985i
\(366\) 0 0
\(367\) 0 0 −0.321439 0.946930i \(-0.604167\pi\)
0.321439 + 0.946930i \(0.395833\pi\)
\(368\) 0 0
\(369\) −0.108761 0.0726721i −0.108761 0.0726721i
\(370\) −0.284338 + 0.492488i −0.284338 + 0.492488i
\(371\) 0 0
\(372\) 0 0
\(373\) −1.71723 0.991445i −1.71723 0.991445i −0.923880 0.382683i \(-0.875000\pi\)
−0.793353 0.608761i \(-0.791667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.869474 + 0.991445i −0.869474 + 0.991445i
\(378\) 0 0
\(379\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.608761 1.79335i 0.608761 1.79335i
\(387\) 0 0
\(388\) −1.29335 1.47479i −1.29335 1.47479i
\(389\) −0.662827 + 1.60021i −0.662827 + 1.60021i 0.130526 + 0.991445i \(0.458333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(393\) 0 0
\(394\) 0.996552 0.491445i 0.996552 0.491445i
\(395\) 0 0
\(396\) 0 0
\(397\) 1.09645 1.25026i 1.09645 1.25026i 0.130526 0.991445i \(-0.458333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.465466 0.357164i −0.465466 0.357164i
\(401\) 1.60876 + 0.793353i 1.60876 + 0.793353i 1.00000 \(0\)
0.608761 + 0.793353i \(0.291667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.184592 0.184592i 0.184592 0.184592i
\(405\) −0.483342 + 0.423880i −0.483342 + 0.423880i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.67303 + 0.448288i −1.67303 + 0.448288i −0.965926 0.258819i \(-0.916667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(410\) 0.0833732 + 0.0109763i 0.0833732 + 0.0109763i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000i 1.00000i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.896873 0.442289i \(-0.854167\pi\)
0.896873 + 0.442289i \(0.145833\pi\)
\(420\) 0 0
\(421\) −1.98289 −1.98289 −0.991445 0.130526i \(-0.958333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0.860919 + 0.860919i 0.860919 + 0.860919i
\(425\) −0.542046 0.224523i −0.542046 0.224523i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.946930 0.321439i \(-0.104167\pi\)
−0.946930 + 0.321439i \(0.895833\pi\)
\(432\) 0 0
\(433\) 0.991445 + 0.130526i 0.991445 + 0.130526i 0.608761 0.793353i \(-0.291667\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.293353 0.257264i −0.293353 0.257264i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(440\) 0 0
\(441\) 1.00000 1.00000
\(442\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0.490985 + 0.0321808i 0.490985 + 0.0321808i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.389345 + 0.0255190i −0.389345 + 0.0255190i −0.258819 0.965926i \(-0.583333\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(450\) −0.224523 + 0.542046i −0.224523 + 0.542046i
\(451\) 0 0
\(452\) −1.42388 + 0.483342i −1.42388 + 0.483342i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.608761 0.793353i 0.608761 0.793353i −0.382683 0.923880i \(-0.625000\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.158919 1.20711i 0.158919 1.20711i −0.707107 0.707107i \(-0.750000\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 1.24871 + 0.423880i 1.24871 + 0.423880i
\(465\) 0 0
\(466\) −0.735499 + 1.49144i −0.735499 + 1.49144i
\(467\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(468\) 0.965926 0.258819i 0.965926 0.258819i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.608761 1.05441i 0.608761 1.05441i
\(478\) 0 0
\(479\) 0 0 0.751840 0.659346i \(-0.229167\pi\)
−0.751840 + 0.659346i \(0.770833\pi\)
\(480\) 0 0
\(481\) −0.665060 0.583242i −0.665060 0.583242i
\(482\) 0.835400 + 1.25026i 0.835400 + 1.25026i
\(483\) 0 0
\(484\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(485\) 0.164600 1.25026i 0.164600 1.25026i
\(486\) 0 0
\(487\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(488\) 0.793353 + 1.60876i 0.793353 + 1.60876i
\(489\) 0 0
\(490\) −0.576581 + 0.284338i −0.576581 + 0.284338i
\(491\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(492\) 0 0
\(493\) 1.31587 + 0.0862466i 1.31587 + 0.0862466i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(500\) −0.0667151 1.01788i −0.0667151 1.01788i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(504\) 0 0
\(505\) 0.167466 + 0.0109763i 0.167466 + 0.0109763i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.965926 0.258819i −0.965926 0.258819i −0.258819 0.965926i \(-0.583333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.923880 0.382683i 0.923880 0.382683i
\(513\) 0 0
\(514\) 0.965926 1.67303i 0.965926 1.67303i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −0.483342 + 0.423880i −0.483342 + 0.423880i
\(521\) 1.65938 1.10876i 1.65938 1.10876i 0.793353 0.608761i \(-0.208333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(522\) 0.0862466 1.31587i 0.0862466 1.31587i
\(523\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.608761 + 0.793353i 0.608761 + 0.793353i
\(530\) −0.0511923 + 0.781044i −0.0511923 + 0.781044i
\(531\) 0 0
\(532\) 0 0
\(533\) −0.0420463 + 0.123864i −0.0420463 + 0.123864i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −0.382683 0.0761205i −0.382683 0.0761205i
\(539\) 0 0
\(540\) 0 0
\(541\) 1.75928 0.349942i 1.75928 0.349942i 0.793353 0.608761i \(-0.208333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0.793353 0.608761i 0.793353 0.608761i
\(545\) 0.250839i 0.250839i
\(546\) 0 0
\(547\) 0 0 0.555570 0.831470i \(-0.312500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(548\) −1.71723 + 0.991445i −1.71723 + 0.991445i
\(549\) 1.34861 1.18270i 1.34861 1.18270i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.57469 + 1.05217i −1.57469 + 1.05217i
\(555\) 0 0
\(556\) 0 0
\(557\) −1.05441 + 0.608761i −1.05441 + 0.608761i −0.923880 0.382683i \(-0.875000\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.130526 0.226078i −0.130526 0.226078i
\(563\) 0 0 0.793353 0.608761i \(-0.208333\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(564\) 0 0
\(565\) −0.837173 0.483342i −0.837173 0.483342i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.465926 0.607206i 0.465926 0.607206i −0.500000 0.866025i \(-0.666667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(570\) 0 0
\(571\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.608761 0.793353i −0.608761 0.793353i
\(577\) 0.860919 0.860919i 0.860919 0.860919i −0.130526 0.991445i \(-0.541667\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(578\) 0.608761 0.793353i 0.608761 0.793353i
\(579\) 0 0
\(580\) 0.324423 + 0.783227i 0.324423 + 0.783227i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.65938 1.10876i 1.65938 1.10876i
\(585\) 0.534534 + 0.357164i 0.534534 + 0.357164i
\(586\) −1.78480 + 0.739288i −1.78480 + 0.739288i
\(587\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −0.284338 + 0.837633i −0.284338 + 0.837633i
\(593\) 1.83195 0.758819i 1.83195 0.758819i 0.866025 0.500000i \(-0.166667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.67303 0.448288i −1.67303 0.448288i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(600\) 0 0
\(601\) −0.0578541 + 0.117317i −0.0578541 + 0.117317i −0.923880 0.382683i \(-0.875000\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.0420463 + 0.641502i 0.0420463 + 0.641502i
\(606\) 0 0
\(607\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −0.441296 + 1.06538i −0.441296 + 1.06538i
\(611\) 0 0
\(612\) −0.793353 0.608761i −0.793353 0.608761i
\(613\) −0.133975 + 0.500000i −0.133975 + 0.500000i 0.866025 + 0.500000i \(0.166667\pi\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.665060 + 1.34861i 0.665060 + 1.34861i 0.923880 + 0.382683i \(0.125000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(618\) 0 0
\(619\) 0 0 −0.555570 0.831470i \(-0.687500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.0488389 0.0488389i 0.0488389 0.0488389i
\(626\) −1.47479 + 1.29335i −1.47479 + 1.29335i
\(627\) 0 0
\(628\) 0.258819 0.448288i 0.258819 0.448288i
\(629\) −0.0578541 + 0.882683i −0.0578541 + 0.882683i
\(630\) 0 0
\(631\) 0 0 −0.991445 0.130526i \(-0.958333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.991445 1.13053i −0.991445 1.13053i
\(635\) 0 0
\(636\) 0 0
\(637\) −0.258819 0.965926i −0.258819 0.965926i
\(638\) 0 0
\(639\) 0 0
\(640\) 0.576581 + 0.284338i 0.576581 + 0.284338i
\(641\) 0.123864 + 0.0420463i 0.123864 + 0.0420463i 0.382683 0.923880i \(-0.375000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(642\) 0 0
\(643\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(649\) 0 0
\(650\) 0.581687 + 0.0765806i 0.581687 + 0.0765806i
\(651\) 0 0
\(652\) 0 0
\(653\) 1.85747 0.630526i 1.85747 0.630526i 0.866025 0.500000i \(-0.166667\pi\)
0.991445 0.130526i \(-0.0416667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.130526 0.00855514i 0.130526 0.00855514i
\(657\) −1.50046 1.31587i −1.50046 1.31587i
\(658\) 0 0
\(659\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(660\) 0 0
\(661\) 0.207107 + 0.158919i 0.207107 + 0.158919i 0.707107 0.707107i \(-0.250000\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.882683 + 0.0578541i 0.882683 + 0.0578541i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.837633 + 0.284338i −0.837633 + 0.284338i −0.707107 0.707107i \(-0.750000\pi\)
−0.130526 + 0.991445i \(0.541667\pi\)
\(674\) 1.88981 0.641502i 1.88981 0.641502i
\(675\) 0 0
\(676\) −0.500000 0.866025i −0.500000 0.866025i
\(677\) −1.38268 0.923880i −1.38268 0.923880i −0.382683 0.923880i \(-0.625000\pi\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.630526 + 0.125419i 0.630526 + 0.125419i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.946930 0.321439i \(-0.895833\pi\)
0.946930 + 0.321439i \(0.104167\pi\)
\(684\) 0 0
\(685\) −1.20711 0.409758i −1.20711 0.409758i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.17604 0.315118i −1.17604 0.315118i
\(690\) 0 0
\(691\) 0 0 −0.321439 0.946930i \(-0.604167\pi\)
0.321439 + 0.946930i \(0.395833\pi\)
\(692\) −0.732626 0.835400i −0.732626 0.835400i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.123864 0.0420463i 0.123864 0.0420463i
\(698\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.0675653 0.513210i 0.0675653 0.513210i
\(707\) 0 0
\(708\) 0 0
\(709\) −0.882683 1.78990i −0.882683 1.78990i −0.500000 0.866025i \(-0.666667\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.758819 0.0999004i 0.758819 0.0999004i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.321439 0.946930i \(-0.395833\pi\)
−0.321439 + 0.946930i \(0.604167\pi\)
\(720\) 0.125419 0.630526i 0.125419 0.630526i
\(721\) 0 0
\(722\) −0.608761 0.793353i −0.608761 0.793353i
\(723\) 0 0
\(724\) −1.24871 + 1.42388i −1.24871 + 1.42388i
\(725\) 0.342192 0.693897i 0.342192 0.693897i
\(726\) 0 0
\(727\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(730\) 1.23929 + 0.332066i 1.23929 + 0.332066i
\(731\) 0 0
\(732\) 0 0
\(733\) 0.241181 0.0999004i 0.241181 0.0999004i −0.258819 0.965926i \(-0.583333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.0420463 0.123864i −0.0420463 0.123864i
\(739\) 0 0 0.991445 0.130526i \(-0.0416667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(740\) −0.525388 + 0.217623i −0.525388 + 0.217623i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.0654031 0.997859i \(-0.479167\pi\)
−0.0654031 + 0.997859i \(0.520833\pi\)
\(744\) 0 0
\(745\) −0.492488 0.998667i −0.492488 0.998667i
\(746\) −0.758819 1.83195i −0.758819 1.83195i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.0654031 0.997859i \(-0.479167\pi\)
−0.0654031 + 0.997859i \(0.520833\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −1.29335 + 0.257264i −1.29335 + 0.257264i
\(755\) 0 0
\(756\) 0 0
\(757\) −0.860919 + 1.12197i −0.860919 + 1.12197i 0.130526 + 0.991445i \(0.458333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.60021 + 0.923880i 1.60021 + 0.923880i 0.991445 + 0.130526i \(0.0416667\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.0420463 0.641502i −0.0420463 0.641502i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.662827 + 0.382683i −0.662827 + 0.382683i −0.793353 0.608761i \(-0.791667\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.57469 1.05217i 1.57469 1.05217i
\(773\) 1.12197 0.860919i 1.12197 0.860919i 0.130526 0.991445i \(-0.458333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.128293 1.95737i −0.128293 1.95737i
\(777\) 0 0
\(778\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.793353 + 0.608761i −0.793353 + 0.608761i
\(785\) 0.326384 0.0649219i 0.326384 0.0649219i
\(786\) 0 0
\(787\) 0 0 0.896873 0.442289i \(-0.145833\pi\)
−0.896873 + 0.442289i \(0.854167\pi\)
\(788\) 1.08979 + 0.216773i 1.08979 + 0.216773i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.49144 0.996552i −1.49144 0.996552i
\(794\) 1.63099 0.324423i 1.63099 0.324423i
\(795\) 0 0
\(796\) 0 0
\(797\) −0.465926 0.607206i −0.465926 0.607206i 0.500000 0.866025i \(-0.333333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.151851 0.566715i −0.151851 0.566715i
\(801\) −0.292893 0.707107i −0.292893 0.707107i
\(802\) 0.793353 + 1.60876i 0.793353 + 1.60876i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.258819 0.0340742i 0.258819 0.0340742i
\(809\) −0.284338 0.837633i −0.284338 0.837633i −0.991445 0.130526i \(-0.958333\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(810\) −0.641502 + 0.0420463i −0.641502 + 0.0420463i
\(811\) 0 0 −0.831470 0.555570i \(-0.812500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.60021 0.662827i −1.60021 0.662827i
\(819\) 0 0
\(820\) 0.0594624 + 0.0594624i 0.0594624 + 0.0594624i
\(821\) 1.10876 + 0.0726721i 1.10876 + 0.0726721i 0.608761 0.793353i \(-0.291667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.195090 0.980785i \(-0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(828\) 0 0
\(829\) 0.500000 1.86603i 0.500000 1.86603i 1.00000i \(-0.5\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(833\) −0.608761 + 0.793353i −0.608761 + 0.793353i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.659346 0.751840i \(-0.270833\pi\)
−0.659346 + 0.751840i \(0.729167\pi\)
\(840\) 0 0
\(841\) −0.0964520 + 0.732626i −0.0964520 + 0.732626i
\(842\) −1.57313 1.20711i −1.57313 1.20711i
\(843\) 0 0
\(844\) 0 0
\(845\) 0.206647 0.608761i 0.206647 0.608761i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.158919 + 1.20711i 0.158919 + 1.20711i
\(849\) 0 0
\(850\) −0.293353 0.508103i −0.293353 0.508103i
\(851\) 0 0
\(852\) 0 0
\(853\) 0.349942 + 1.75928i 0.349942 + 1.75928i 0.608761 + 0.793353i \(0.291667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.85747 0.369474i −1.85747 0.369474i −0.866025 0.500000i \(-0.833333\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(858\) 0 0
\(859\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0.0932386 0.708218i 0.0932386 0.708218i
\(866\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.0761205 0.382683i −0.0761205 0.382683i
\(873\) −1.85747 + 0.630526i −1.85747 + 0.630526i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.31587 0.0862466i 1.31587 0.0862466i 0.608761 0.793353i \(-0.291667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.78990 + 0.117317i 1.78990 + 0.117317i 0.923880 0.382683i \(-0.125000\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(882\) 0.793353 + 0.608761i 0.793353 + 0.608761i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0.369934 + 0.324423i 0.369934 + 0.324423i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.324423 0.216773i −0.324423 0.216773i
\(899\) 0 0
\(900\) −0.508103 + 0.293353i −0.508103 + 0.293353i
\(901\) 0.465926 + 1.12484i 0.465926 + 1.12484i
\(902\) 0 0
\(903\) 0 0
\(904\) −1.42388 0.483342i −1.42388 0.483342i
\(905\) −1.21752 −1.21752
\(906\) 0 0
\(907\) 0 0 −0.896873 0.442289i \(-0.854167\pi\)
0.896873 + 0.442289i \(0.145833\pi\)
\(908\) 0 0
\(909\) −0.0999004 0.241181i −0.0999004 0.241181i
\(910\) 0 0
\(911\) 0 0 −0.980785 0.195090i \(-0.937500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.965926 0.258819i 0.965926 0.258819i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.860919 0.860919i 0.860919 0.860919i
\(923\) 0 0
\(924\) 0 0
\(925\) 0.465466 + 0.229542i 0.465466 + 0.229542i
\(926\) 0 0
\(927\) 0 0
\(928\) 0.732626 + 1.09645i 0.732626 + 1.09645i
\(929\) 0.869474 0.991445i 0.869474 0.991445i −0.130526 0.991445i \(-0.541667\pi\)
1.00000 \(0\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.49144 + 0.735499i −1.49144 + 0.735499i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.923880 + 0.382683i 0.923880 + 0.382683i
\(937\) −0.0999004 + 0.241181i −0.0999004 + 0.241181i −0.965926 0.258819i \(-0.916667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.382683 + 1.92388i −0.382683 + 1.92388i 1.00000i \(0.5\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.997859 0.0654031i \(-0.979167\pi\)
0.997859 + 0.0654031i \(0.0208333\pi\)
\(948\) 0 0
\(949\) −0.882683 + 1.78990i −0.882683 + 1.78990i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.60021 0.923880i −1.60021 0.923880i −0.991445 0.130526i \(-0.958333\pi\)
−0.608761 0.793353i \(-0.708333\pi\)
\(954\) 1.12484 0.465926i 1.12484 0.465926i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(962\) −0.172572 0.867580i −0.172572 0.867580i
\(963\) 0 0
\(964\) −0.0983454 + 1.50046i −0.0983454 + 1.50046i
\(965\) 1.17604 + 0.315118i 1.17604 + 0.315118i
\(966\) 0 0
\(967\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(968\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(969\) 0 0
\(970\) 0.891699 0.891699i 0.891699 0.891699i
\(971\) 0 0 −0.608761 0.793353i \(-0.708333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −0.349942 + 1.75928i −0.349942 + 1.75928i
\(977\) −0.207107 1.57313i −0.207107 1.57313i −0.707107 0.707107i \(-0.750000\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.630526 0.125419i −0.630526 0.125419i
\(981\) −0.349942 + 0.172572i −0.349942 + 0.172572i
\(982\) 0 0
\(983\) 0 0 0.980785 0.195090i \(-0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(984\) 0 0
\(985\) 0.357164 + 0.618627i 0.357164 + 0.618627i
\(986\) 0.991445 + 0.869474i 0.991445 + 0.869474i
\(987\) 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 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.130526 + 1.99144i 0.130526 + 1.99144i 0.130526 + 0.991445i \(0.458333\pi\)
1.00000i \(0.5\pi\)
\(998\) 0 0
\(999\) 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 884.1.ct.a.687.1 yes 16
4.3 odd 2 CM 884.1.ct.a.687.1 yes 16
13.6 odd 12 884.1.cn.a.279.1 16
17.5 odd 16 884.1.cn.a.583.1 yes 16
52.19 even 12 884.1.cn.a.279.1 16
68.39 even 16 884.1.cn.a.583.1 yes 16
221.175 even 48 inner 884.1.ct.a.175.1 yes 16
884.175 odd 48 inner 884.1.ct.a.175.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
884.1.cn.a.279.1 16 13.6 odd 12
884.1.cn.a.279.1 16 52.19 even 12
884.1.cn.a.583.1 yes 16 17.5 odd 16
884.1.cn.a.583.1 yes 16 68.39 even 16
884.1.ct.a.175.1 yes 16 221.175 even 48 inner
884.1.ct.a.175.1 yes 16 884.175 odd 48 inner
884.1.ct.a.687.1 yes 16 1.1 even 1 trivial
884.1.ct.a.687.1 yes 16 4.3 odd 2 CM