Properties

Label 548.1.k.a.211.1
Level $548$
Weight $1$
Character 548.211
Analytic conductor $0.273$
Analytic rank $0$
Dimension $16$
Projective image $D_{17}$
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] = [548,1,Mod(59,548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(548, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([17, 26]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("548.59");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 548 = 2^{2} \cdot 137 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 548.k (of order \(34\), 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: \(0.273487626923\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{17}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{17} - \cdots)\)

Embedding invariants

Embedding label 211.1
Root \(0.850217 + 0.526432i\) of defining polynomial
Character \(\chi\) \(=\) 548.211
Dual form 548.1.k.a.187.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.739009 + 0.673696i) q^{2} +(0.0922684 + 0.995734i) q^{4} +(-1.45285 + 1.32445i) q^{5} +(-0.602635 + 0.798017i) q^{8} +(0.445738 - 0.895163i) q^{9} +O(q^{10})\) \(q+(0.739009 + 0.673696i) q^{2} +(0.0922684 + 0.995734i) q^{4} +(-1.45285 + 1.32445i) q^{5} +(-0.602635 + 0.798017i) q^{8} +(0.445738 - 0.895163i) q^{9} -1.96595 q^{10} +(-0.0505009 + 0.177492i) q^{13} +(-0.982973 + 0.183750i) q^{16} +(0.726337 + 0.961826i) q^{17} +(0.932472 - 0.361242i) q^{18} +(-1.45285 - 1.32445i) q^{20} +(0.264344 - 2.85272i) q^{25} +(-0.156896 + 0.0971461i) q^{26} +(0.538007 - 0.100571i) q^{29} +(-0.850217 - 0.526432i) q^{32} +(-0.111208 + 1.20013i) q^{34} +(0.932472 + 0.361242i) q^{36} +1.47802 q^{37} +(-0.181395 - 1.95756i) q^{40} +1.86494 q^{41} +(0.538007 + 1.89090i) q^{45} +(-0.850217 - 0.526432i) q^{49} +(2.11722 - 1.93010i) q^{50} +(-0.181395 - 0.0339085i) q^{52} +(-1.12388 + 0.435393i) q^{53} +(0.465346 + 0.288130i) q^{58} +(-0.876298 - 1.75984i) q^{61} +(-0.273663 - 0.961826i) q^{64} +(-0.161709 - 0.324756i) q^{65} +(-0.890705 + 0.811985i) q^{68} +(0.445738 + 0.895163i) q^{72} +(-0.243964 - 0.857445i) q^{73} +(1.09227 + 0.995734i) q^{74} +(1.18475 - 1.56886i) q^{80} +(-0.602635 - 0.798017i) q^{81} +(1.37821 + 1.25640i) q^{82} +(-2.32915 - 0.435393i) q^{85} +(-1.25664 + 1.14558i) q^{89} +(-0.876298 + 1.75984i) q^{90} +(0.184537 + 1.99147i) q^{97} +(-0.273663 - 0.961826i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - q^{4} - 2 q^{5} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{2} - q^{4} - 2 q^{5} - q^{8} - q^{9} - 2 q^{10} - 2 q^{13} - q^{16} + 15 q^{17} - q^{18} - 2 q^{20} - 3 q^{25} - 2 q^{26} - 2 q^{29} - q^{32} - 2 q^{34} - q^{36} - 2 q^{37} - 2 q^{40} - 2 q^{41} - 2 q^{45} - q^{49} - 3 q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{61} - q^{64} - 4 q^{65} - 2 q^{68} - q^{72} - 2 q^{73} + 15 q^{74} - 2 q^{80} - q^{81} - 2 q^{82} - 4 q^{85} - 2 q^{89} - 2 q^{90} - 2 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/548\mathbb{Z}\right)^\times\).

\(n\) \(275\) \(277\)
\(\chi(n)\) \(-1\) \(e\left(\frac{14}{17}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.739009 + 0.673696i 0.739009 + 0.673696i
\(3\) 0 0 0.850217 0.526432i \(-0.176471\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(4\) 0.0922684 + 0.995734i 0.0922684 + 0.995734i
\(5\) −1.45285 + 1.32445i −1.45285 + 1.32445i −0.602635 + 0.798017i \(0.705882\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(6\) 0 0
\(7\) 0 0 0.273663 0.961826i \(-0.411765\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(8\) −0.602635 + 0.798017i −0.602635 + 0.798017i
\(9\) 0.445738 0.895163i 0.445738 0.895163i
\(10\) −1.96595 −1.96595
\(11\) 0 0 −0.0922684 0.995734i \(-0.529412\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(12\) 0 0
\(13\) −0.0505009 + 0.177492i −0.0505009 + 0.177492i −0.982973 0.183750i \(-0.941176\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.982973 + 0.183750i −0.982973 + 0.183750i
\(17\) 0.726337 + 0.961826i 0.726337 + 0.961826i 1.00000 \(0\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(18\) 0.932472 0.361242i 0.932472 0.361242i
\(19\) 0 0 0.932472 0.361242i \(-0.117647\pi\)
−0.932472 + 0.361242i \(0.882353\pi\)
\(20\) −1.45285 1.32445i −1.45285 1.32445i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.982973 0.183750i \(-0.0588235\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(24\) 0 0
\(25\) 0.264344 2.85272i 0.264344 2.85272i
\(26\) −0.156896 + 0.0971461i −0.156896 + 0.0971461i
\(27\) 0 0
\(28\) 0 0
\(29\) 0.538007 0.100571i 0.538007 0.100571i 0.0922684 0.995734i \(-0.470588\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(30\) 0 0
\(31\) 0 0 −0.932472 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(32\) −0.850217 0.526432i −0.850217 0.526432i
\(33\) 0 0
\(34\) −0.111208 + 1.20013i −0.111208 + 1.20013i
\(35\) 0 0
\(36\) 0.932472 + 0.361242i 0.932472 + 0.361242i
\(37\) 1.47802 1.47802 0.739009 0.673696i \(-0.235294\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.181395 1.95756i −0.181395 1.95756i
\(41\) 1.86494 1.86494 0.932472 0.361242i \(-0.117647\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(42\) 0 0
\(43\) 0 0 0.932472 0.361242i \(-0.117647\pi\)
−0.932472 + 0.361242i \(0.882353\pi\)
\(44\) 0 0
\(45\) 0.538007 + 1.89090i 0.538007 + 1.89090i
\(46\) 0 0
\(47\) 0 0 0.445738 0.895163i \(-0.352941\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(48\) 0 0
\(49\) −0.850217 0.526432i −0.850217 0.526432i
\(50\) 2.11722 1.93010i 2.11722 1.93010i
\(51\) 0 0
\(52\) −0.181395 0.0339085i −0.181395 0.0339085i
\(53\) −1.12388 + 0.435393i −1.12388 + 0.435393i −0.850217 0.526432i \(-0.823529\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.465346 + 0.288130i 0.465346 + 0.288130i
\(59\) 0 0 0.445738 0.895163i \(-0.352941\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(60\) 0 0
\(61\) −0.876298 1.75984i −0.876298 1.75984i −0.602635 0.798017i \(-0.705882\pi\)
−0.273663 0.961826i \(-0.588235\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.273663 0.961826i −0.273663 0.961826i
\(65\) −0.161709 0.324756i −0.161709 0.324756i
\(66\) 0 0
\(67\) 0 0 0.273663 0.961826i \(-0.411765\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(68\) −0.890705 + 0.811985i −0.890705 + 0.811985i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.0922684 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(72\) 0.445738 + 0.895163i 0.445738 + 0.895163i
\(73\) −0.243964 0.857445i −0.243964 0.857445i −0.982973 0.183750i \(-0.941176\pi\)
0.739009 0.673696i \(-0.235294\pi\)
\(74\) 1.09227 + 0.995734i 1.09227 + 0.995734i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.850217 0.526432i \(-0.823529\pi\)
0.850217 + 0.526432i \(0.176471\pi\)
\(80\) 1.18475 1.56886i 1.18475 1.56886i
\(81\) −0.602635 0.798017i −0.602635 0.798017i
\(82\) 1.37821 + 1.25640i 1.37821 + 1.25640i
\(83\) 0 0 0.602635 0.798017i \(-0.294118\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(84\) 0 0
\(85\) −2.32915 0.435393i −2.32915 0.435393i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.25664 + 1.14558i −1.25664 + 1.14558i −0.273663 + 0.961826i \(0.588235\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(90\) −0.876298 + 1.75984i −0.876298 + 1.75984i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.184537 + 1.99147i 0.184537 + 1.99147i 0.0922684 + 0.995734i \(0.470588\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(98\) −0.273663 0.961826i −0.273663 0.961826i
\(99\) 0 0
\(100\) 2.86494 2.86494
\(101\) −1.58561 0.614268i −1.58561 0.614268i −0.602635 0.798017i \(-0.705882\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(102\) 0 0
\(103\) 0 0 0.0922684 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(104\) −0.111208 0.147263i −0.111208 0.147263i
\(105\) 0 0
\(106\) −1.12388 0.435393i −1.12388 0.435393i
\(107\) 0 0 0.850217 0.526432i \(-0.176471\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(108\) 0 0
\(109\) 1.67148 + 0.312454i 1.67148 + 0.312454i 0.932472 0.361242i \(-0.117647\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0.0822551 0.887674i 0.0822551 0.887674i −0.850217 0.526432i \(-0.823529\pi\)
0.932472 0.361242i \(-0.117647\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.149783 + 0.526432i 0.149783 + 0.526432i
\(117\) 0.136374 + 0.124322i 0.136374 + 0.124322i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.982973 + 0.183750i −0.982973 + 0.183750i
\(122\) 0.538007 1.89090i 0.538007 1.89090i
\(123\) 0 0
\(124\) 0 0
\(125\) 2.20949 + 2.92583i 2.20949 + 2.92583i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0.445738 0.895163i 0.445738 0.895163i
\(129\) 0 0
\(130\) 0.0992820 0.348940i 0.0992820 0.348940i
\(131\) 0 0 −0.982973 0.183750i \(-0.941176\pi\)
0.982973 + 0.183750i \(0.0588235\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −1.20527 −1.20527
\(137\) −0.273663 0.961826i −0.273663 0.961826i
\(138\) 0 0
\(139\) 0 0 −0.739009 0.673696i \(-0.764706\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.273663 + 0.961826i −0.273663 + 0.961826i
\(145\) −0.648443 + 0.858677i −0.648443 + 0.858677i
\(146\) 0.397365 0.798017i 0.397365 0.798017i
\(147\) 0 0
\(148\) 0.136374 + 1.47171i 0.136374 + 1.47171i
\(149\) −0.890705 1.17948i −0.890705 1.17948i −0.982973 0.183750i \(-0.941176\pi\)
0.0922684 0.995734i \(-0.470588\pi\)
\(150\) 0 0
\(151\) 0 0 0.850217 0.526432i \(-0.176471\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(152\) 0 0
\(153\) 1.18475 0.221468i 1.18475 0.221468i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.09227 + 0.995734i 1.09227 + 0.995734i 1.00000 \(0\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.93247 0.361242i 1.93247 0.361242i
\(161\) 0 0
\(162\) 0.0922684 0.995734i 0.0922684 0.995734i
\(163\) 0 0 0.850217 0.526432i \(-0.176471\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(164\) 0.172075 + 1.85699i 0.172075 + 1.85699i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.850217 0.526432i \(-0.176471\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(168\) 0 0
\(169\) 0.821264 + 0.508505i 0.821264 + 0.508505i
\(170\) −1.42794 1.89090i −1.42794 1.89090i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.510366 0.197717i −0.510366 0.197717i 0.0922684 0.995734i \(-0.470588\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.70043 −1.70043
\(179\) 0 0 0.445738 0.895163i \(-0.352941\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(180\) −1.83319 + 0.710182i −1.83319 + 0.710182i
\(181\) 0.538007 0.100571i 0.538007 0.100571i 0.0922684 0.995734i \(-0.470588\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.14734 + 1.95756i −2.14734 + 1.95756i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.602635 0.798017i \(-0.294118\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(192\) 0 0
\(193\) −0.537235 0.711414i −0.537235 0.711414i 0.445738 0.895163i \(-0.352941\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(194\) −1.20527 + 1.59603i −1.20527 + 1.59603i
\(195\) 0 0
\(196\) 0.445738 0.895163i 0.445738 0.895163i
\(197\) 1.67148 + 0.312454i 1.67148 + 0.312454i 0.932472 0.361242i \(-0.117647\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(198\) 0 0
\(199\) 0 0 −0.445738 0.895163i \(-0.647059\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(200\) 2.11722 + 1.93010i 2.11722 + 1.93010i
\(201\) 0 0
\(202\) −0.757949 1.52217i −0.757949 1.52217i
\(203\) 0 0
\(204\) 0 0
\(205\) −2.70949 + 2.47002i −2.70949 + 2.47002i
\(206\) 0 0
\(207\) 0 0
\(208\) 0.0170269 0.183750i 0.0170269 0.183750i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.739009 0.673696i \(-0.764706\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(212\) −0.537235 1.07891i −0.537235 1.07891i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.02474 + 1.35698i 1.02474 + 1.35698i
\(219\) 0 0
\(220\) 0 0
\(221\) −0.207397 + 0.0803461i −0.207397 + 0.0803461i
\(222\) 0 0
\(223\) 0 0 −0.932472 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(224\) 0 0
\(225\) −2.43582 1.50820i −2.43582 1.50820i
\(226\) 0.658809 0.600584i 0.658809 0.600584i
\(227\) 0 0 0.445738 0.895163i \(-0.352941\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(228\) 0 0
\(229\) −0.510366 1.79375i −0.510366 1.79375i −0.602635 0.798017i \(-0.705882\pi\)
0.0922684 0.995734i \(-0.470588\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.243964 + 0.489946i −0.243964 + 0.489946i
\(233\) −1.20527 −1.20527 −0.602635 0.798017i \(-0.705882\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(234\) 0.0170269 + 0.183750i 0.0170269 + 0.183750i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.445738 0.895163i \(-0.647059\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(240\) 0 0
\(241\) −0.537235 0.711414i −0.537235 0.711414i 0.445738 0.895163i \(-0.352941\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(242\) −0.850217 0.526432i −0.850217 0.526432i
\(243\) 0 0
\(244\) 1.67148 1.03494i 1.67148 1.03494i
\(245\) 1.93247 0.361242i 1.93247 0.361242i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.338291 + 3.65074i −0.338291 + 3.65074i
\(251\) 0 0 0.982973 0.183750i \(-0.0588235\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.932472 0.361242i 0.932472 0.361242i
\(257\) −0.111208 0.147263i −0.111208 0.147263i 0.739009 0.673696i \(-0.235294\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.308450 0.190984i 0.308450 0.190984i
\(261\) 0.149783 0.526432i 0.149783 0.526432i
\(262\) 0 0
\(263\) 0 0 −0.0922684 0.995734i \(-0.529412\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(264\) 0 0
\(265\) 1.05617 2.12108i 1.05617 2.12108i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.658809 0.600584i 0.658809 0.600584i −0.273663 0.961826i \(-0.588235\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(270\) 0 0
\(271\) 0 0 0.850217 0.526432i \(-0.176471\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(272\) −0.890705 0.811985i −0.890705 0.811985i
\(273\) 0 0
\(274\) 0.445738 0.895163i 0.445738 0.895163i
\(275\) 0 0
\(276\) 0 0
\(277\) −1.70043 + 1.05286i −1.70043 + 1.05286i −0.850217 + 0.526432i \(0.823529\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.404479 + 1.42160i −0.404479 + 1.42160i 0.445738 + 0.895163i \(0.352941\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(282\) 0 0
\(283\) 0 0 0.445738 0.895163i \(-0.352941\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.850217 + 0.526432i −0.850217 + 0.526432i
\(289\) −0.123880 + 0.435393i −0.123880 + 0.435393i
\(290\) −1.05769 + 0.197717i −1.05769 + 0.197717i
\(291\) 0 0
\(292\) 0.831277 0.322039i 0.831277 0.322039i
\(293\) 0.172075 0.0666624i 0.172075 0.0666624i −0.273663 0.961826i \(-0.588235\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.890705 + 1.17948i −0.890705 + 1.17948i
\(297\) 0 0
\(298\) 0.136374 1.47171i 0.136374 1.47171i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.60395 + 1.39618i 3.60395 + 1.39618i
\(306\) 1.02474 + 0.634493i 1.02474 + 0.634493i
\(307\) 0 0 −0.602635 0.798017i \(-0.705882\pi\)
0.602635 + 0.798017i \(0.294118\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0.538007 + 1.89090i 0.538007 + 1.89090i 0.445738 + 0.895163i \(0.352941\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(314\) 0.136374 + 1.47171i 0.136374 + 1.47171i
\(315\) 0 0
\(316\) 0 0
\(317\) 1.73901 0.673696i 1.73901 0.673696i 0.739009 0.673696i \(-0.235294\pi\)
1.00000 \(0\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.67148 + 1.03494i 1.67148 + 1.03494i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.739009 0.673696i 0.739009 0.673696i
\(325\) 0.492986 + 0.190984i 0.492986 + 0.190984i
\(326\) 0 0
\(327\) 0 0
\(328\) −1.12388 + 1.48826i −1.12388 + 1.48826i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.602635 0.798017i \(-0.294118\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(332\) 0 0
\(333\) 0.658809 1.32307i 0.658809 1.32307i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.136374 + 0.124322i 0.136374 + 0.124322i 0.739009 0.673696i \(-0.235294\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(338\) 0.264344 + 0.929072i 0.264344 + 0.929072i
\(339\) 0 0
\(340\) 0.218629 2.35939i 0.218629 2.35939i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.243964 0.489946i −0.243964 0.489946i
\(347\) 0 0 −0.273663 0.961826i \(-0.588235\pi\)
0.273663 + 0.961826i \(0.411765\pi\)
\(348\) 0 0
\(349\) −0.537235 1.07891i −0.537235 1.07891i −0.982973 0.183750i \(-0.941176\pi\)
0.445738 0.895163i \(-0.352941\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.757949 0.469302i −0.757949 0.469302i 0.0922684 0.995734i \(-0.470588\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.25664 1.14558i −1.25664 1.14558i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.982973 0.183750i \(-0.941176\pi\)
0.982973 + 0.183750i \(0.0588235\pi\)
\(360\) −1.83319 0.710182i −1.83319 0.710182i
\(361\) 0.739009 0.673696i 0.739009 0.673696i
\(362\) 0.465346 + 0.288130i 0.465346 + 0.288130i
\(363\) 0 0
\(364\) 0 0
\(365\) 1.49009 + 0.922623i 1.49009 + 0.922623i
\(366\) 0 0
\(367\) 0 0 0.982973 0.183750i \(-0.0588235\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(368\) 0 0
\(369\) 0.831277 1.66943i 0.831277 1.66943i
\(370\) −2.90570 −2.90570
\(371\) 0 0
\(372\) 0 0
\(373\) 0.172075 + 0.0666624i 0.172075 + 0.0666624i 0.445738 0.895163i \(-0.352941\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.00931926 + 0.100571i −0.00931926 + 0.100571i
\(378\) 0 0
\(379\) 0 0 −0.850217 0.526432i \(-0.823529\pi\)
0.850217 + 0.526432i \(0.176471\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.982973 0.183750i \(-0.941176\pi\)
0.982973 + 0.183750i \(0.0588235\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.0822551 0.887674i 0.0822551 0.887674i
\(387\) 0 0
\(388\) −1.96595 + 0.367499i −1.96595 + 0.367499i
\(389\) −0.537235 + 0.711414i −0.537235 + 0.711414i −0.982973 0.183750i \(-0.941176\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.932472 0.361242i 0.932472 0.361242i
\(393\) 0 0
\(394\) 1.02474 + 1.35698i 1.02474 + 1.35698i
\(395\) 0 0
\(396\) 0 0
\(397\) −1.25664 + 0.778076i −1.25664 + 0.778076i −0.982973 0.183750i \(-0.941176\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.264344 + 2.85272i 0.264344 + 2.85272i
\(401\) −0.547326 −0.547326 −0.273663 0.961826i \(-0.588235\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.465346 1.63552i 0.465346 1.63552i
\(405\) 1.93247 + 0.361242i 1.93247 + 0.361242i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.37821 + 1.25640i 1.37821 + 1.25640i 0.932472 + 0.361242i \(0.117647\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(410\) −3.66638 −3.66638
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.136374 0.124322i 0.136374 0.124322i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.602635 0.798017i \(-0.294118\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(420\) 0 0
\(421\) −1.70043 −1.70043 −0.850217 0.526432i \(-0.823529\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0.329838 1.15926i 0.329838 1.15926i
\(425\) 2.93582 1.81779i 2.93582 1.81779i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.739009 0.673696i \(-0.764706\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(432\) 0 0
\(433\) 1.02474 1.35698i 1.02474 1.35698i 0.0922684 0.995734i \(-0.470588\pi\)
0.932472 0.361242i \(-0.117647\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.156896 + 1.69318i −0.156896 + 1.69318i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.982973 0.183750i \(-0.941176\pi\)
0.982973 + 0.183750i \(0.0588235\pi\)
\(440\) 0 0
\(441\) −0.850217 + 0.526432i −0.850217 + 0.526432i
\(442\) −0.207397 0.0803461i −0.207397 0.0803461i
\(443\) 0 0 −0.850217 0.526432i \(-0.823529\pi\)
0.850217 + 0.526432i \(0.176471\pi\)
\(444\) 0 0
\(445\) 0.308450 3.32870i 0.308450 3.32870i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.510366 0.197717i −0.510366 0.197717i 0.0922684 0.995734i \(-0.470588\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(450\) −0.784029 2.75558i −0.784029 2.75558i
\(451\) 0 0
\(452\) 0.891477 0.891477
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.44574 + 0.895163i 1.44574 + 0.895163i 1.00000 \(0\)
0.445738 + 0.895163i \(0.352941\pi\)
\(458\) 0.831277 1.66943i 0.831277 1.66943i
\(459\) 0 0
\(460\) 0 0
\(461\) 0.658809 0.600584i 0.658809 0.600584i −0.273663 0.961826i \(-0.588235\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(462\) 0 0
\(463\) 0 0 −0.982973 0.183750i \(-0.941176\pi\)
0.982973 + 0.183750i \(0.0588235\pi\)
\(464\) −0.510366 + 0.197717i −0.510366 + 0.197717i
\(465\) 0 0
\(466\) −0.890705 0.811985i −0.890705 0.811985i
\(467\) 0 0 −0.602635 0.798017i \(-0.705882\pi\)
0.602635 + 0.798017i \(0.294118\pi\)
\(468\) −0.111208 + 0.147263i −0.111208 + 0.147263i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.111208 + 1.20013i −0.111208 + 1.20013i
\(478\) 0 0
\(479\) 0 0 0.739009 0.673696i \(-0.235294\pi\)
−0.739009 + 0.673696i \(0.764706\pi\)
\(480\) 0 0
\(481\) −0.0746412 + 0.262337i −0.0746412 + 0.262337i
\(482\) 0.0822551 0.887674i 0.0822551 0.887674i
\(483\) 0 0
\(484\) −0.273663 0.961826i −0.273663 0.961826i
\(485\) −2.90570 2.64890i −2.90570 2.64890i
\(486\) 0 0
\(487\) 0 0 −0.445738 0.895163i \(-0.647059\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(488\) 1.93247 + 0.361242i 1.93247 + 0.361242i
\(489\) 0 0
\(490\) 1.67148 + 1.03494i 1.67148 + 1.03494i
\(491\) 0 0 0.602635 0.798017i \(-0.294118\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(492\) 0 0
\(493\) 0.487506 + 0.444420i 0.487506 + 0.444420i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.850217 0.526432i \(-0.823529\pi\)
0.850217 + 0.526432i \(0.176471\pi\)
\(500\) −2.70949 + 2.47002i −2.70949 + 2.47002i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.273663 0.961826i \(-0.588235\pi\)
0.273663 + 0.961826i \(0.411765\pi\)
\(504\) 0 0
\(505\) 3.11722 1.20762i 3.11722 1.20762i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.404479 1.42160i −0.404479 1.42160i −0.850217 0.526432i \(-0.823529\pi\)
0.445738 0.895163i \(-0.352941\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.932472 + 0.361242i 0.932472 + 0.361242i
\(513\) 0 0
\(514\) 0.0170269 0.183750i 0.0170269 0.183750i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0.356612 + 0.0666624i 0.356612 + 0.0666624i
\(521\) −0.0505009 0.544991i −0.0505009 0.544991i −0.982973 0.183750i \(-0.941176\pi\)
0.932472 0.361242i \(-0.117647\pi\)
\(522\) 0.465346 0.288130i 0.465346 0.288130i
\(523\) 0 0 0.0922684 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.932472 0.361242i 0.932472 0.361242i
\(530\) 2.20949 0.855960i 2.20949 0.855960i
\(531\) 0 0
\(532\) 0 0
\(533\) −0.0941813 + 0.331013i −0.0941813 + 0.331013i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.891477 0.891477
\(539\) 0 0
\(540\) 0 0
\(541\) −0.510366 + 1.79375i −0.510366 + 1.79375i 0.0922684 + 0.995734i \(0.470588\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.111208 1.20013i −0.111208 1.20013i
\(545\) −2.84224 + 1.75984i −2.84224 + 1.75984i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0.932472 0.361242i 0.932472 0.361242i
\(549\) −1.96595 −1.96595
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.96595 0.367499i −1.96595 0.367499i
\(555\) 0 0
\(556\) 0 0
\(557\) −0.537235 + 1.07891i −0.537235 + 1.07891i 0.445738 + 0.895163i \(0.352941\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.25664 + 0.778076i −1.25664 + 0.778076i
\(563\) 0 0 0.273663 0.961826i \(-0.411765\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(564\) 0 0
\(565\) 1.05617 + 1.39860i 1.05617 + 1.39860i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.510366 1.79375i −0.510366 1.79375i −0.602635 0.798017i \(-0.705882\pi\)
0.0922684 0.995734i \(-0.470588\pi\)
\(570\) 0 0
\(571\) 0 0 0.982973 0.183750i \(-0.0588235\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.982973 0.183750i −0.982973 0.183750i
\(577\) −1.45285 + 0.271585i −1.45285 + 0.271585i −0.850217 0.526432i \(-0.823529\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(578\) −0.384871 + 0.238302i −0.384871 + 0.238302i
\(579\) 0 0
\(580\) −0.914845 0.566448i −0.914845 0.566448i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0.831277 + 0.322039i 0.831277 + 0.322039i
\(585\) −0.362789 −0.362789
\(586\) 0.172075 + 0.0666624i 0.172075 + 0.0666624i
\(587\) 0 0 −0.273663 0.961826i \(-0.588235\pi\)
0.273663 + 0.961826i \(0.411765\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.45285 + 0.271585i −1.45285 + 0.271585i
\(593\) 0.149783 + 0.526432i 0.149783 + 0.526432i 1.00000 \(0\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.09227 0.995734i 1.09227 0.995734i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.932472 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(600\) 0 0
\(601\) −1.58561 + 0.614268i −1.58561 + 0.614268i −0.982973 0.183750i \(-0.941176\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.18475 1.56886i 1.18475 1.56886i
\(606\) 0 0
\(607\) 0 0 0.445738 0.895163i \(-0.352941\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 1.72275 + 3.45976i 1.72275 + 3.45976i
\(611\) 0 0
\(612\) 0.329838 + 1.15926i 0.329838 + 1.15926i
\(613\) 0.831277 + 1.66943i 0.831277 + 1.66943i 0.739009 + 0.673696i \(0.235294\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.890705 + 0.811985i −0.890705 + 0.811985i −0.982973 0.183750i \(-0.941176\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(618\) 0 0
\(619\) 0 0 0.0922684 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4.26902 0.798017i −4.26902 0.798017i
\(626\) −0.876298 + 1.75984i −0.876298 + 1.75984i
\(627\) 0 0
\(628\) −0.890705 + 1.17948i −0.890705 + 1.17948i
\(629\) 1.07354 + 1.42160i 1.07354 + 1.42160i
\(630\) 0 0
\(631\) 0 0 0.602635 0.798017i \(-0.294118\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 1.73901 + 0.673696i 1.73901 + 0.673696i
\(635\) 0 0
\(636\) 0 0
\(637\) 0.136374 0.124322i 0.136374 0.124322i
\(638\) 0 0
\(639\) 0 0
\(640\) 0.538007 + 1.89090i 0.538007 + 1.89090i
\(641\) 1.18475 0.221468i 1.18475 0.221468i 0.445738 0.895163i \(-0.352941\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(642\) 0 0
\(643\) 0 0 0.445738 0.895163i \(-0.352941\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.932472 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(648\) 1.00000 1.00000
\(649\) 0 0
\(650\) 0.235656 + 0.473262i 0.235656 + 0.473262i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.757949 0.469302i −0.757949 0.469302i 0.0922684 0.995734i \(-0.470588\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.83319 + 0.342683i −1.83319 + 0.342683i
\(657\) −0.876298 0.163808i −0.876298 0.163808i
\(658\) 0 0
\(659\) 0 0 0.850217 0.526432i \(-0.176471\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(660\) 0 0
\(661\) −0.0505009 + 0.544991i −0.0505009 + 0.544991i 0.932472 + 0.361242i \(0.117647\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.37821 0.533922i 1.37821 0.533922i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.726337 + 0.961826i 0.726337 + 0.961826i 1.00000 \(0\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(674\) 0.0170269 + 0.183750i 0.0170269 + 0.183750i
\(675\) 0 0
\(676\) −0.430559 + 0.864680i −0.430559 + 0.864680i
\(677\) −0.890705 + 1.17948i −0.890705 + 1.17948i 0.0922684 + 0.995734i \(0.470588\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.75108 1.59632i 1.75108 1.59632i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.739009 0.673696i \(-0.764706\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(684\) 0 0
\(685\) 1.67148 + 1.03494i 1.67148 + 1.03494i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.0205220 0.221468i −0.0205220 0.221468i
\(690\) 0 0
\(691\) 0 0 −0.982973 0.183750i \(-0.941176\pi\)
0.982973 + 0.183750i \(0.0588235\pi\)
\(692\) 0.149783 0.526432i 0.149783 0.526432i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.35458 + 1.79375i 1.35458 + 1.79375i
\(698\) 0.329838 1.15926i 0.329838 1.15926i
\(699\) 0 0
\(700\) 0 0
\(701\) −0.181395 + 0.0339085i −0.181395 + 0.0339085i −0.273663 0.961826i \(-0.588235\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −0.243964 0.857445i −0.243964 0.857445i
\(707\) 0 0
\(708\) 0 0
\(709\) 0.0822551 0.887674i 0.0822551 0.887674i −0.850217 0.526432i \(-0.823529\pi\)
0.932472 0.361242i \(-0.117647\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.156896 1.69318i −0.156896 1.69318i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.0922684 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(720\) −0.876298 1.75984i −0.876298 1.75984i
\(721\) 0 0
\(722\) 1.00000 1.00000
\(723\) 0 0
\(724\) 0.149783 + 0.526432i 0.149783 + 0.526432i
\(725\) −0.144682 1.56137i −0.144682 1.56137i
\(726\) 0 0
\(727\) 0 0 0.445738 0.895163i \(-0.352941\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(728\) 0 0
\(729\) −0.982973 + 0.183750i −0.982973 + 0.183750i
\(730\) 0.479620 + 1.68569i 0.479620 + 1.68569i
\(731\) 0 0
\(732\) 0 0
\(733\) 0.136374 0.124322i 0.136374 0.124322i −0.602635 0.798017i \(-0.705882\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 1.73901 0.673696i 1.73901 0.673696i
\(739\) 0 0 0.602635 0.798017i \(-0.294118\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(740\) −2.14734 1.95756i −2.14734 1.95756i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.850217 0.526432i \(-0.823529\pi\)
0.850217 + 0.526432i \(0.176471\pi\)
\(744\) 0 0
\(745\) 2.85623 + 0.533922i 2.85623 + 0.533922i
\(746\) 0.0822551 + 0.165190i 0.0822551 + 0.165190i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.0922684 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −0.0746412 + 0.0680444i −0.0746412 + 0.0680444i
\(755\) 0 0
\(756\) 0 0
\(757\) 0.658809 + 1.32307i 0.658809 + 1.32307i 0.932472 + 0.361242i \(0.117647\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.831277 + 1.66943i 0.831277 + 1.66943i 0.739009 + 0.673696i \(0.235294\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.42794 + 1.89090i −1.42794 + 1.89090i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.172075 0.0666624i 0.172075 0.0666624i −0.273663 0.961826i \(-0.588235\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.658809 0.600584i 0.658809 0.600584i
\(773\) −1.58561 0.981767i −1.58561 0.981767i −0.982973 0.183750i \(-0.941176\pi\)
−0.602635 0.798017i \(-0.705882\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.70043 1.05286i −1.70043 1.05286i
\(777\) 0 0
\(778\) −0.876298 + 0.163808i −0.876298 + 0.163808i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.932472 + 0.361242i 0.932472 + 0.361242i
\(785\) −2.90570 −2.90570
\(786\) 0 0
\(787\) 0 0 −0.445738 0.895163i \(-0.647059\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(788\) −0.156896 + 1.69318i −0.156896 + 1.69318i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.356612 0.0666624i 0.356612 0.0666624i
\(794\) −1.45285 0.271585i −1.45285 0.271585i
\(795\) 0 0
\(796\) 0 0
\(797\) 0.172075 1.85699i 0.172075 1.85699i −0.273663 0.961826i \(-0.588235\pi\)
0.445738 0.895163i \(-0.352941\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.72651 + 2.28628i −1.72651 + 2.28628i
\(801\) 0.465346 + 1.63552i 0.465346 + 1.63552i
\(802\) −0.404479 0.368731i −0.404479 0.368731i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 1.44574 0.895163i 1.44574 0.895163i
\(809\) −0.510366 + 1.79375i −0.510366 + 1.79375i 0.0922684 + 0.995734i \(0.470588\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(810\) 1.18475 + 1.56886i 1.18475 + 1.56886i
\(811\) 0 0 −0.0922684 0.995734i \(-0.529412\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0.172075 + 1.85699i 0.172075 + 1.85699i
\(819\) 0 0
\(820\) −2.70949 2.47002i −2.70949 2.47002i
\(821\) 0.184537 0.184537 0.0922684 0.995734i \(-0.470588\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.739009 0.673696i \(-0.235294\pi\)
−0.739009 + 0.673696i \(0.764706\pi\)
\(828\) 0 0
\(829\) 0.538007 1.89090i 0.538007 1.89090i 0.0922684 0.995734i \(-0.470588\pi\)
0.445738 0.895163i \(-0.352941\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.184537 0.184537
\(833\) −0.111208 1.20013i −0.111208 1.20013i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.602635 0.798017i \(-0.705882\pi\)
0.602635 + 0.798017i \(0.294118\pi\)
\(840\) 0 0
\(841\) −0.653136 + 0.253026i −0.653136 + 0.253026i
\(842\) −1.25664 1.14558i −1.25664 1.14558i
\(843\) 0 0
\(844\) 0 0
\(845\) −1.86666 + 0.348940i −1.86666 + 0.348940i
\(846\) 0 0
\(847\) 0 0
\(848\) 1.02474 0.634493i 1.02474 0.634493i
\(849\) 0 0
\(850\) 3.39424 + 0.634493i 3.39424 + 0.634493i
\(851\) 0 0
\(852\) 0 0
\(853\) −1.58561 0.614268i −1.58561 0.614268i −0.602635 0.798017i \(-0.705882\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.243964 0.489946i −0.243964 0.489946i 0.739009 0.673696i \(-0.235294\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 1.00335 0.388701i 1.00335 0.388701i
\(866\) 1.67148 0.312454i 1.67148 0.312454i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −1.25664 + 1.14558i −1.25664 + 1.14558i
\(873\) 1.86494 + 0.722483i 1.86494 + 0.722483i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.658809 + 0.600584i 0.658809 + 0.600584i 0.932472 0.361242i \(-0.117647\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.831277 1.66943i 0.831277 1.66943i 0.0922684 0.995734i \(-0.470588\pi\)
0.739009 0.673696i \(-0.235294\pi\)
\(882\) −0.982973 0.183750i −0.982973 0.183750i
\(883\) 0 0 −0.445738 0.895163i \(-0.647059\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(884\) −0.0991395 0.199099i −0.0991395 0.199099i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.445738 0.895163i \(-0.647059\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.47048 2.25214i 2.47048 2.25214i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.243964 0.489946i −0.243964 0.489946i
\(899\) 0 0
\(900\) 1.27702 2.56459i 1.27702 2.56459i
\(901\) −1.23509 0.764734i −1.23509 0.764734i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.658809 + 0.600584i 0.658809 + 0.600584i
\(905\) −0.648443 + 0.858677i −0.648443 + 0.858677i
\(906\) 0 0
\(907\) 0 0 −0.982973 0.183750i \(-0.941176\pi\)
0.982973 + 0.183750i \(0.0588235\pi\)
\(908\) 0 0
\(909\) −1.25664 + 1.14558i −1.25664 + 1.14558i
\(910\) 0 0
\(911\) 0 0 0.739009 0.673696i \(-0.235294\pi\)
−0.739009 + 0.673696i \(0.764706\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.465346 + 1.63552i 0.465346 + 1.63552i
\(915\) 0 0
\(916\) 1.73901 0.673696i 1.73901 0.673696i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.0922684 0.995734i \(-0.529412\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.891477 0.891477
\(923\) 0 0
\(924\) 0 0
\(925\) 0.390705 4.21638i 0.390705 4.21638i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.510366 0.197717i −0.510366 0.197717i
\(929\) −0.156896 + 0.0971461i −0.156896 + 0.0971461i −0.602635 0.798017i \(-0.705882\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.111208 1.20013i −0.111208 1.20013i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −0.181395 + 0.0339085i −0.181395 + 0.0339085i
\(937\) 0.329838 0.436776i 0.329838 0.436776i −0.602635 0.798017i \(-0.705882\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.12388 + 0.435393i −1.12388 + 0.435393i −0.850217 0.526432i \(-0.823529\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.602635 0.798017i \(-0.705882\pi\)
0.602635 + 0.798017i \(0.294118\pi\)
\(948\) 0 0
\(949\) 0.164510 0.164510
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.45285 0.271585i −1.45285 0.271585i −0.602635 0.798017i \(-0.705882\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(954\) −0.890705 + 0.811985i −0.890705 + 0.811985i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.739009 + 0.673696i 0.739009 + 0.673696i
\(962\) −0.231896 + 0.143584i −0.231896 + 0.143584i
\(963\) 0 0
\(964\) 0.658809 0.600584i 0.658809 0.600584i
\(965\) 1.72275 + 0.322039i 1.72275 + 0.322039i
\(966\) 0 0
\(967\) 0 0 0.602635 0.798017i \(-0.294118\pi\)
−0.602635 + 0.798017i \(0.705882\pi\)
\(968\) 0.445738 0.895163i 0.445738 0.895163i
\(969\) 0 0
\(970\) −0.362789 3.91512i −0.362789 3.91512i
\(971\) 0 0 −0.602635 0.798017i \(-0.705882\pi\)
0.602635 + 0.798017i \(0.294118\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.18475 + 1.56886i 1.18475 + 1.56886i
\(977\) −0.510366 + 0.197717i −0.510366 + 0.197717i −0.602635 0.798017i \(-0.705882\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.538007 + 1.89090i 0.538007 + 1.89090i
\(981\) 1.02474 1.35698i 1.02474 1.35698i
\(982\) 0 0
\(983\) 0 0 0.0922684 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(984\) 0 0
\(985\) −2.84224 + 1.75984i −2.84224 + 1.75984i
\(986\) 0.0608671 + 0.656861i 0.0608671 + 0.656861i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.850217 0.526432i \(-0.823529\pi\)
0.850217 + 0.526432i \(0.176471\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.12388 0.435393i −1.12388 0.435393i −0.273663 0.961826i \(-0.588235\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(998\) 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 548.1.k.a.211.1 yes 16
4.3 odd 2 CM 548.1.k.a.211.1 yes 16
137.50 even 17 inner 548.1.k.a.187.1 16
548.187 odd 34 inner 548.1.k.a.187.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
548.1.k.a.187.1 16 137.50 even 17 inner
548.1.k.a.187.1 16 548.187 odd 34 inner
548.1.k.a.211.1 yes 16 1.1 even 1 trivial
548.1.k.a.211.1 yes 16 4.3 odd 2 CM