Properties

Label 756.2.bo.b.85.1
Level $756$
Weight $2$
Character 756.85
Analytic conductor $6.037$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,2,Mod(85,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.bo (of order \(9\), degree \(6\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 85.1
Character \(\chi\) \(=\) 756.85
Dual form 756.2.bo.b.169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.72800 + 0.118316i) q^{3} +(0.626647 - 3.55389i) q^{5} +(-0.766044 + 0.642788i) q^{7} +(2.97200 - 0.408902i) q^{9} +O(q^{10})\) \(q+(-1.72800 + 0.118316i) q^{3} +(0.626647 - 3.55389i) q^{5} +(-0.766044 + 0.642788i) q^{7} +(2.97200 - 0.408902i) q^{9} +(0.0433566 + 0.245888i) q^{11} +(3.04704 - 1.10903i) q^{13} +(-0.662366 + 6.21529i) q^{15} +(1.13833 - 1.97165i) q^{17} +(1.63940 + 2.83953i) q^{19} +(1.24768 - 1.20138i) q^{21} +(-7.07760 - 5.93881i) q^{23} +(-7.53901 - 2.74398i) q^{25} +(-5.08726 + 1.05822i) q^{27} +(-4.13113 - 1.50361i) q^{29} +(-3.83329 - 3.21651i) q^{31} +(-0.104013 - 0.419765i) q^{33} +(1.80436 + 3.12524i) q^{35} +(0.771841 - 1.33687i) q^{37} +(-5.13409 + 2.27693i) q^{39} +(1.07512 - 0.391311i) q^{41} +(0.401383 + 2.27636i) q^{43} +(0.409203 - 10.8184i) q^{45} +(8.59159 - 7.20920i) q^{47} +(0.173648 - 0.984808i) q^{49} +(-1.73376 + 3.54170i) q^{51} -5.74802 q^{53} +0.901028 q^{55} +(-3.16886 - 4.71276i) q^{57} +(1.74022 - 9.86927i) q^{59} +(-4.88170 + 4.09623i) q^{61} +(-2.01385 + 2.22360i) q^{63} +(-2.03196 - 11.5238i) q^{65} +(-10.8172 + 3.93713i) q^{67} +(12.9328 + 9.42490i) q^{69} +(-0.402666 + 0.697438i) q^{71} +(-6.95480 - 12.0461i) q^{73} +(13.3521 + 3.84962i) q^{75} +(-0.191267 - 0.160492i) q^{77} +(-7.28350 - 2.65098i) q^{79} +(8.66560 - 2.43052i) q^{81} +(5.51788 + 2.00834i) q^{83} +(-6.29369 - 5.28103i) q^{85} +(7.31651 + 2.10946i) q^{87} +(7.12272 + 12.3369i) q^{89} +(-1.62130 + 2.80817i) q^{91} +(7.00450 + 5.10460i) q^{93} +(11.1187 - 4.04688i) q^{95} +(0.0151777 + 0.0860770i) q^{97} +(0.229400 + 0.713050i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 54 q + 3 q^{3} - 3 q^{5} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 54 q + 3 q^{3} - 3 q^{5} - 9 q^{9} + 9 q^{13} + 3 q^{15} + 3 q^{21} + 12 q^{23} + 27 q^{25} + 39 q^{27} + 30 q^{29} - 9 q^{31} - 6 q^{33} + 6 q^{35} - 18 q^{39} - 27 q^{41} + 9 q^{43} - 81 q^{45} + 9 q^{47} + 12 q^{53} - 21 q^{57} - 24 q^{59} - 3 q^{63} - 78 q^{65} - 9 q^{67} + 6 q^{69} - 30 q^{71} + 57 q^{75} + 9 q^{77} + 99 q^{81} + 78 q^{83} + 18 q^{85} + 21 q^{87} + 12 q^{89} - 51 q^{93} - 36 q^{95} + 36 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\) \(1\) \(1\)

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 0
\(3\) −1.72800 + 0.118316i −0.997664 + 0.0683099i
\(4\) 0 0
\(5\) 0.626647 3.55389i 0.280245 1.58935i −0.441546 0.897239i \(-0.645570\pi\)
0.721791 0.692111i \(-0.243319\pi\)
\(6\) 0 0
\(7\) −0.766044 + 0.642788i −0.289538 + 0.242951i
\(8\) 0 0
\(9\) 2.97200 0.408902i 0.990668 0.136301i
\(10\) 0 0
\(11\) 0.0433566 + 0.245888i 0.0130725 + 0.0741379i 0.990646 0.136456i \(-0.0435713\pi\)
−0.977574 + 0.210594i \(0.932460\pi\)
\(12\) 0 0
\(13\) 3.04704 1.10903i 0.845098 0.307590i 0.117058 0.993125i \(-0.462654\pi\)
0.728040 + 0.685535i \(0.240432\pi\)
\(14\) 0 0
\(15\) −0.662366 + 6.21529i −0.171022 + 1.60478i
\(16\) 0 0
\(17\) 1.13833 1.97165i 0.276086 0.478194i −0.694323 0.719664i \(-0.744296\pi\)
0.970408 + 0.241469i \(0.0776293\pi\)
\(18\) 0 0
\(19\) 1.63940 + 2.83953i 0.376105 + 0.651433i 0.990492 0.137572i \(-0.0439298\pi\)
−0.614387 + 0.789005i \(0.710596\pi\)
\(20\) 0 0
\(21\) 1.24768 1.20138i 0.272265 0.262162i
\(22\) 0 0
\(23\) −7.07760 5.93881i −1.47578 1.23833i −0.910552 0.413395i \(-0.864343\pi\)
−0.565231 0.824933i \(-0.691213\pi\)
\(24\) 0 0
\(25\) −7.53901 2.74398i −1.50780 0.548795i
\(26\) 0 0
\(27\) −5.08726 + 1.05822i −0.979043 + 0.203655i
\(28\) 0 0
\(29\) −4.13113 1.50361i −0.767131 0.279213i −0.0713350 0.997452i \(-0.522726\pi\)
−0.695796 + 0.718240i \(0.744948\pi\)
\(30\) 0 0
\(31\) −3.83329 3.21651i −0.688479 0.577702i 0.229991 0.973193i \(-0.426130\pi\)
−0.918470 + 0.395491i \(0.870575\pi\)
\(32\) 0 0
\(33\) −0.104013 0.419765i −0.0181063 0.0730718i
\(34\) 0 0
\(35\) 1.80436 + 3.12524i 0.304992 + 0.528262i
\(36\) 0 0
\(37\) 0.771841 1.33687i 0.126890 0.219780i −0.795580 0.605848i \(-0.792834\pi\)
0.922470 + 0.386069i \(0.126167\pi\)
\(38\) 0 0
\(39\) −5.13409 + 2.27693i −0.822112 + 0.364600i
\(40\) 0 0
\(41\) 1.07512 0.391311i 0.167905 0.0611125i −0.256700 0.966491i \(-0.582635\pi\)
0.424605 + 0.905379i \(0.360413\pi\)
\(42\) 0 0
\(43\) 0.401383 + 2.27636i 0.0612103 + 0.347141i 0.999997 + 0.00264062i \(0.000840538\pi\)
−0.938786 + 0.344500i \(0.888048\pi\)
\(44\) 0 0
\(45\) 0.409203 10.8184i 0.0610004 1.61271i
\(46\) 0 0
\(47\) 8.59159 7.20920i 1.25321 1.05157i 0.256841 0.966454i \(-0.417318\pi\)
0.996371 0.0851161i \(-0.0271261\pi\)
\(48\) 0 0
\(49\) 0.173648 0.984808i 0.0248069 0.140687i
\(50\) 0 0
\(51\) −1.73376 + 3.54170i −0.242775 + 0.495937i
\(52\) 0 0
\(53\) −5.74802 −0.789551 −0.394775 0.918778i \(-0.629178\pi\)
−0.394775 + 0.918778i \(0.629178\pi\)
\(54\) 0 0
\(55\) 0.901028 0.121495
\(56\) 0 0
\(57\) −3.16886 4.71276i −0.419726 0.624220i
\(58\) 0 0
\(59\) 1.74022 9.86927i 0.226557 1.28487i −0.633129 0.774047i \(-0.718230\pi\)
0.859686 0.510823i \(-0.170659\pi\)
\(60\) 0 0
\(61\) −4.88170 + 4.09623i −0.625038 + 0.524469i −0.899383 0.437163i \(-0.855983\pi\)
0.274345 + 0.961631i \(0.411539\pi\)
\(62\) 0 0
\(63\) −2.01385 + 2.22360i −0.253721 + 0.280148i
\(64\) 0 0
\(65\) −2.03196 11.5238i −0.252034 1.42936i
\(66\) 0 0
\(67\) −10.8172 + 3.93713i −1.32153 + 0.480997i −0.903948 0.427642i \(-0.859344\pi\)
−0.417581 + 0.908640i \(0.637122\pi\)
\(68\) 0 0
\(69\) 12.9328 + 9.42490i 1.55693 + 1.13463i
\(70\) 0 0
\(71\) −0.402666 + 0.697438i −0.0477877 + 0.0827706i −0.888930 0.458043i \(-0.848550\pi\)
0.841142 + 0.540814i \(0.181884\pi\)
\(72\) 0 0
\(73\) −6.95480 12.0461i −0.813997 1.40988i −0.910046 0.414508i \(-0.863954\pi\)
0.0960482 0.995377i \(-0.469380\pi\)
\(74\) 0 0
\(75\) 13.3521 + 3.84962i 1.54177 + 0.444515i
\(76\) 0 0
\(77\) −0.191267 0.160492i −0.0217969 0.0182897i
\(78\) 0 0
\(79\) −7.28350 2.65098i −0.819458 0.298258i −0.101933 0.994791i \(-0.532503\pi\)
−0.717525 + 0.696533i \(0.754725\pi\)
\(80\) 0 0
\(81\) 8.66560 2.43052i 0.962844 0.270057i
\(82\) 0 0
\(83\) 5.51788 + 2.00834i 0.605666 + 0.220444i 0.626606 0.779336i \(-0.284444\pi\)
−0.0209400 + 0.999781i \(0.506666\pi\)
\(84\) 0 0
\(85\) −6.29369 5.28103i −0.682646 0.572808i
\(86\) 0 0
\(87\) 7.31651 + 2.10946i 0.784412 + 0.226158i
\(88\) 0 0
\(89\) 7.12272 + 12.3369i 0.755007 + 1.30771i 0.945371 + 0.325997i \(0.105700\pi\)
−0.190364 + 0.981714i \(0.560967\pi\)
\(90\) 0 0
\(91\) −1.62130 + 2.80817i −0.169958 + 0.294376i
\(92\) 0 0
\(93\) 7.00450 + 5.10460i 0.726333 + 0.529323i
\(94\) 0 0
\(95\) 11.1187 4.04688i 1.14076 0.415201i
\(96\) 0 0
\(97\) 0.0151777 + 0.0860770i 0.00154106 + 0.00873980i 0.985569 0.169277i \(-0.0541431\pi\)
−0.984028 + 0.178016i \(0.943032\pi\)
\(98\) 0 0
\(99\) 0.229400 + 0.713050i 0.0230556 + 0.0716642i
\(100\) 0 0
\(101\) −3.77493 + 3.16755i −0.375620 + 0.315183i −0.810980 0.585074i \(-0.801066\pi\)
0.435360 + 0.900256i \(0.356621\pi\)
\(102\) 0 0
\(103\) 0.931621 5.28348i 0.0917953 0.520597i −0.903887 0.427771i \(-0.859299\pi\)
0.995682 0.0928259i \(-0.0295900\pi\)
\(104\) 0 0
\(105\) −3.48771 5.18695i −0.340365 0.506194i
\(106\) 0 0
\(107\) 5.79919 0.560629 0.280315 0.959908i \(-0.409561\pi\)
0.280315 + 0.959908i \(0.409561\pi\)
\(108\) 0 0
\(109\) 2.08236 0.199454 0.0997269 0.995015i \(-0.468203\pi\)
0.0997269 + 0.995015i \(0.468203\pi\)
\(110\) 0 0
\(111\) −1.17557 + 2.40144i −0.111580 + 0.227934i
\(112\) 0 0
\(113\) 1.01369 5.74890i 0.0953595 0.540811i −0.899277 0.437379i \(-0.855907\pi\)
0.994637 0.103431i \(-0.0329822\pi\)
\(114\) 0 0
\(115\) −25.5411 + 21.4315i −2.38172 + 1.99850i
\(116\) 0 0
\(117\) 8.60233 4.54199i 0.795286 0.419907i
\(118\) 0 0
\(119\) 0.395338 + 2.24207i 0.0362406 + 0.205530i
\(120\) 0 0
\(121\) 10.2780 3.74090i 0.934367 0.340082i
\(122\) 0 0
\(123\) −1.81151 + 0.803391i −0.163338 + 0.0724394i
\(124\) 0 0
\(125\) −5.45430 + 9.44713i −0.487848 + 0.844977i
\(126\) 0 0
\(127\) −4.35273 7.53914i −0.386242 0.668991i 0.605699 0.795694i \(-0.292894\pi\)
−0.991941 + 0.126703i \(0.959560\pi\)
\(128\) 0 0
\(129\) −0.962921 3.88606i −0.0847805 0.342149i
\(130\) 0 0
\(131\) 15.3371 + 12.8694i 1.34001 + 1.12440i 0.981623 + 0.190831i \(0.0611182\pi\)
0.358389 + 0.933572i \(0.383326\pi\)
\(132\) 0 0
\(133\) −3.08107 1.12142i −0.267163 0.0972393i
\(134\) 0 0
\(135\) 0.572890 + 18.7427i 0.0493065 + 1.61311i
\(136\) 0 0
\(137\) 4.39528 + 1.59975i 0.375514 + 0.136676i 0.522880 0.852406i \(-0.324857\pi\)
−0.147367 + 0.989082i \(0.547080\pi\)
\(138\) 0 0
\(139\) 15.2156 + 12.7674i 1.29057 + 1.08292i 0.991693 + 0.128627i \(0.0410571\pi\)
0.298879 + 0.954291i \(0.403387\pi\)
\(140\) 0 0
\(141\) −13.9933 + 13.4741i −1.17845 + 1.13472i
\(142\) 0 0
\(143\) 0.404807 + 0.701146i 0.0338517 + 0.0586328i
\(144\) 0 0
\(145\) −7.93242 + 13.7394i −0.658752 + 1.14099i
\(146\) 0 0
\(147\) −0.183546 + 1.72230i −0.0151386 + 0.142053i
\(148\) 0 0
\(149\) −9.71750 + 3.53688i −0.796088 + 0.289752i −0.707864 0.706348i \(-0.750341\pi\)
−0.0882237 + 0.996101i \(0.528119\pi\)
\(150\) 0 0
\(151\) 1.70081 + 9.64575i 0.138410 + 0.784960i 0.972424 + 0.233218i \(0.0749256\pi\)
−0.834015 + 0.551742i \(0.813963\pi\)
\(152\) 0 0
\(153\) 2.57691 6.32520i 0.208331 0.511362i
\(154\) 0 0
\(155\) −13.8333 + 11.6075i −1.11111 + 0.932335i
\(156\) 0 0
\(157\) −3.20583 + 18.1812i −0.255853 + 1.45101i 0.538021 + 0.842932i \(0.319172\pi\)
−0.793874 + 0.608083i \(0.791939\pi\)
\(158\) 0 0
\(159\) 9.93260 0.680084i 0.787706 0.0539341i
\(160\) 0 0
\(161\) 9.23915 0.728147
\(162\) 0 0
\(163\) 20.9694 1.64245 0.821224 0.570606i \(-0.193292\pi\)
0.821224 + 0.570606i \(0.193292\pi\)
\(164\) 0 0
\(165\) −1.55698 + 0.106606i −0.121211 + 0.00829928i
\(166\) 0 0
\(167\) 4.17919 23.7014i 0.323396 1.83407i −0.197321 0.980339i \(-0.563224\pi\)
0.520717 0.853730i \(-0.325665\pi\)
\(168\) 0 0
\(169\) −1.90406 + 1.59770i −0.146466 + 0.122900i
\(170\) 0 0
\(171\) 6.03340 + 7.76874i 0.461386 + 0.594090i
\(172\) 0 0
\(173\) −0.720844 4.08811i −0.0548048 0.310813i 0.945066 0.326879i \(-0.105997\pi\)
−0.999871 + 0.0160658i \(0.994886\pi\)
\(174\) 0 0
\(175\) 7.53901 2.74398i 0.569896 0.207425i
\(176\) 0 0
\(177\) −1.83941 + 17.2600i −0.138259 + 1.29734i
\(178\) 0 0
\(179\) −1.31889 + 2.28438i −0.0985784 + 0.170743i −0.911096 0.412193i \(-0.864763\pi\)
0.812518 + 0.582936i \(0.198096\pi\)
\(180\) 0 0
\(181\) −12.4966 21.6447i −0.928864 1.60884i −0.785226 0.619209i \(-0.787453\pi\)
−0.143638 0.989630i \(-0.545880\pi\)
\(182\) 0 0
\(183\) 7.95095 7.65590i 0.587751 0.565940i
\(184\) 0 0
\(185\) −4.26741 3.58079i −0.313747 0.263265i
\(186\) 0 0
\(187\) 0.534157 + 0.194417i 0.0390615 + 0.0142172i
\(188\) 0 0
\(189\) 3.21685 4.08067i 0.233992 0.296825i
\(190\) 0 0
\(191\) −10.3350 3.76164i −0.747816 0.272183i −0.0601293 0.998191i \(-0.519151\pi\)
−0.687686 + 0.726008i \(0.741374\pi\)
\(192\) 0 0
\(193\) 15.0179 + 12.6015i 1.08101 + 0.907079i 0.996005 0.0893011i \(-0.0284633\pi\)
0.0850100 + 0.996380i \(0.472908\pi\)
\(194\) 0 0
\(195\) 4.87470 + 19.6728i 0.349085 + 1.40880i
\(196\) 0 0
\(197\) −1.24646 2.15893i −0.0888065 0.153817i 0.818200 0.574933i \(-0.194972\pi\)
−0.907007 + 0.421116i \(0.861639\pi\)
\(198\) 0 0
\(199\) −9.41974 + 16.3155i −0.667748 + 1.15657i 0.310784 + 0.950480i \(0.399408\pi\)
−0.978532 + 0.206093i \(0.933925\pi\)
\(200\) 0 0
\(201\) 18.2263 8.08323i 1.28559 0.570147i
\(202\) 0 0
\(203\) 4.13113 1.50361i 0.289948 0.105533i
\(204\) 0 0
\(205\) −0.716958 4.06607i −0.0500745 0.283987i
\(206\) 0 0
\(207\) −23.4630 14.7561i −1.63079 1.02562i
\(208\) 0 0
\(209\) −0.627127 + 0.526222i −0.0433793 + 0.0363995i
\(210\) 0 0
\(211\) 1.77828 10.0851i 0.122422 0.694288i −0.860384 0.509646i \(-0.829776\pi\)
0.982806 0.184642i \(-0.0591126\pi\)
\(212\) 0 0
\(213\) 0.613291 1.25282i 0.0420220 0.0858417i
\(214\) 0 0
\(215\) 8.34145 0.568882
\(216\) 0 0
\(217\) 5.00400 0.339694
\(218\) 0 0
\(219\) 13.4432 + 19.9928i 0.908405 + 1.35099i
\(220\) 0 0
\(221\) 1.28192 7.27013i 0.0862313 0.489042i
\(222\) 0 0
\(223\) 16.4134 13.7724i 1.09912 0.922271i 0.101755 0.994810i \(-0.467554\pi\)
0.997366 + 0.0725381i \(0.0231099\pi\)
\(224\) 0 0
\(225\) −23.5280 5.07238i −1.56853 0.338159i
\(226\) 0 0
\(227\) −2.07688 11.7785i −0.137847 0.781770i −0.972835 0.231501i \(-0.925636\pi\)
0.834988 0.550269i \(-0.185475\pi\)
\(228\) 0 0
\(229\) 6.29311 2.29051i 0.415861 0.151361i −0.125612 0.992079i \(-0.540089\pi\)
0.541472 + 0.840719i \(0.317867\pi\)
\(230\) 0 0
\(231\) 0.349499 + 0.254701i 0.0229953 + 0.0167581i
\(232\) 0 0
\(233\) 9.41192 16.3019i 0.616595 1.06797i −0.373507 0.927627i \(-0.621845\pi\)
0.990102 0.140347i \(-0.0448219\pi\)
\(234\) 0 0
\(235\) −20.2368 35.0512i −1.32011 2.28649i
\(236\) 0 0
\(237\) 12.8996 + 3.71915i 0.837918 + 0.241585i
\(238\) 0 0
\(239\) 11.2766 + 9.46220i 0.729423 + 0.612059i 0.929974 0.367625i \(-0.119829\pi\)
−0.200551 + 0.979683i \(0.564273\pi\)
\(240\) 0 0
\(241\) 19.1442 + 6.96790i 1.23318 + 0.448842i 0.874687 0.484689i \(-0.161067\pi\)
0.358497 + 0.933531i \(0.383289\pi\)
\(242\) 0 0
\(243\) −14.6866 + 5.22522i −0.942148 + 0.335198i
\(244\) 0 0
\(245\) −3.39109 1.23425i −0.216649 0.0788536i
\(246\) 0 0
\(247\) 8.14447 + 6.83402i 0.518220 + 0.434838i
\(248\) 0 0
\(249\) −9.77254 2.81757i −0.619309 0.178556i
\(250\) 0 0
\(251\) −9.83032 17.0266i −0.620484 1.07471i −0.989396 0.145246i \(-0.953603\pi\)
0.368911 0.929465i \(-0.379731\pi\)
\(252\) 0 0
\(253\) 1.15342 1.99778i 0.0725149 0.125599i
\(254\) 0 0
\(255\) 11.5004 + 8.38100i 0.720180 + 0.524839i
\(256\) 0 0
\(257\) −3.99454 + 1.45390i −0.249173 + 0.0906915i −0.463587 0.886051i \(-0.653438\pi\)
0.214414 + 0.976743i \(0.431216\pi\)
\(258\) 0 0
\(259\) 0.268058 + 1.52023i 0.0166563 + 0.0944625i
\(260\) 0 0
\(261\) −12.8925 2.77950i −0.798029 0.172047i
\(262\) 0 0
\(263\) −15.7630 + 13.2267i −0.971986 + 0.815593i −0.982861 0.184348i \(-0.940983\pi\)
0.0108754 + 0.999941i \(0.496538\pi\)
\(264\) 0 0
\(265\) −3.60198 + 20.4278i −0.221268 + 1.25487i
\(266\) 0 0
\(267\) −13.7678 20.4755i −0.842573 1.25308i
\(268\) 0 0
\(269\) 2.76876 0.168814 0.0844071 0.996431i \(-0.473100\pi\)
0.0844071 + 0.996431i \(0.473100\pi\)
\(270\) 0 0
\(271\) 4.61817 0.280534 0.140267 0.990114i \(-0.455204\pi\)
0.140267 + 0.990114i \(0.455204\pi\)
\(272\) 0 0
\(273\) 2.46936 5.04436i 0.149452 0.305298i
\(274\) 0 0
\(275\) 0.347844 1.97272i 0.0209758 0.118959i
\(276\) 0 0
\(277\) 22.0851 18.5316i 1.32697 1.11346i 0.342190 0.939631i \(-0.388831\pi\)
0.984776 0.173827i \(-0.0556132\pi\)
\(278\) 0 0
\(279\) −12.7078 7.99203i −0.760795 0.478471i
\(280\) 0 0
\(281\) −5.64631 32.0218i −0.336831 1.91026i −0.408355 0.912823i \(-0.633898\pi\)
0.0715245 0.997439i \(-0.477214\pi\)
\(282\) 0 0
\(283\) −18.4624 + 6.71976i −1.09748 + 0.399448i −0.826385 0.563106i \(-0.809606\pi\)
−0.271090 + 0.962554i \(0.587384\pi\)
\(284\) 0 0
\(285\) −18.7344 + 8.30856i −1.10973 + 0.492157i
\(286\) 0 0
\(287\) −0.572058 + 0.990834i −0.0337675 + 0.0584871i
\(288\) 0 0
\(289\) 5.90841 + 10.2337i 0.347553 + 0.601980i
\(290\) 0 0
\(291\) −0.0364115 0.146946i −0.00213448 0.00861411i
\(292\) 0 0
\(293\) 19.9015 + 16.6993i 1.16266 + 0.975586i 0.999939 0.0110863i \(-0.00352895\pi\)
0.162720 + 0.986672i \(0.447973\pi\)
\(294\) 0 0
\(295\) −33.9838 12.3691i −1.97862 0.720157i
\(296\) 0 0
\(297\) −0.480770 1.20501i −0.0278971 0.0699219i
\(298\) 0 0
\(299\) −28.1521 10.2465i −1.62808 0.592572i
\(300\) 0 0
\(301\) −1.77069 1.48579i −0.102061 0.0856393i
\(302\) 0 0
\(303\) 6.14833 5.92017i 0.353212 0.340105i
\(304\) 0 0
\(305\) 11.4985 + 19.9159i 0.658401 + 1.14038i
\(306\) 0 0
\(307\) 14.1875 24.5735i 0.809725 1.40249i −0.103329 0.994647i \(-0.532949\pi\)
0.913054 0.407838i \(-0.133717\pi\)
\(308\) 0 0
\(309\) −0.984723 + 9.24011i −0.0560190 + 0.525652i
\(310\) 0 0
\(311\) 18.7881 6.83832i 1.06538 0.387765i 0.250931 0.968005i \(-0.419263\pi\)
0.814446 + 0.580240i \(0.197041\pi\)
\(312\) 0 0
\(313\) 4.54499 + 25.7759i 0.256898 + 1.45694i 0.791153 + 0.611618i \(0.209481\pi\)
−0.534255 + 0.845323i \(0.679408\pi\)
\(314\) 0 0
\(315\) 6.64048 + 8.55042i 0.374149 + 0.481762i
\(316\) 0 0
\(317\) −6.27508 + 5.26542i −0.352444 + 0.295735i −0.801770 0.597632i \(-0.796108\pi\)
0.449327 + 0.893367i \(0.351664\pi\)
\(318\) 0 0
\(319\) 0.190607 1.08098i 0.0106719 0.0605235i
\(320\) 0 0
\(321\) −10.0210 + 0.686139i −0.559320 + 0.0382965i
\(322\) 0 0
\(323\) 7.46473 0.415349
\(324\) 0 0
\(325\) −26.0148 −1.44304
\(326\) 0 0
\(327\) −3.59833 + 0.246377i −0.198988 + 0.0136247i
\(328\) 0 0
\(329\) −1.94756 + 11.0451i −0.107372 + 0.608938i
\(330\) 0 0
\(331\) −25.6482 + 21.5214i −1.40975 + 1.18292i −0.453185 + 0.891416i \(0.649712\pi\)
−0.956568 + 0.291508i \(0.905843\pi\)
\(332\) 0 0
\(333\) 1.74727 4.28878i 0.0957496 0.235024i
\(334\) 0 0
\(335\) 7.21359 + 40.9103i 0.394121 + 2.23517i
\(336\) 0 0
\(337\) 23.0160 8.37713i 1.25376 0.456332i 0.372090 0.928197i \(-0.378641\pi\)
0.881671 + 0.471865i \(0.156419\pi\)
\(338\) 0 0
\(339\) −1.07147 + 10.0541i −0.0581940 + 0.546061i
\(340\) 0 0
\(341\) 0.624702 1.08201i 0.0338295 0.0585944i
\(342\) 0 0
\(343\) 0.500000 + 0.866025i 0.0269975 + 0.0467610i
\(344\) 0 0
\(345\) 41.5994 40.0557i 2.23964 2.15652i
\(346\) 0 0
\(347\) 2.74891 + 2.30661i 0.147569 + 0.123825i 0.713584 0.700570i \(-0.247071\pi\)
−0.566014 + 0.824395i \(0.691515\pi\)
\(348\) 0 0
\(349\) 31.2343 + 11.3684i 1.67193 + 0.608534i 0.992170 0.124898i \(-0.0398603\pi\)
0.679763 + 0.733432i \(0.262082\pi\)
\(350\) 0 0
\(351\) −14.3275 + 8.86638i −0.764745 + 0.473252i
\(352\) 0 0
\(353\) −32.7203 11.9092i −1.74153 0.633864i −0.742186 0.670194i \(-0.766211\pi\)
−0.999340 + 0.0363300i \(0.988433\pi\)
\(354\) 0 0
\(355\) 2.22629 + 1.86808i 0.118159 + 0.0991474i
\(356\) 0 0
\(357\) −0.948419 3.82754i −0.0501957 0.202575i
\(358\) 0 0
\(359\) 3.78864 + 6.56211i 0.199957 + 0.346335i 0.948514 0.316735i \(-0.102587\pi\)
−0.748558 + 0.663070i \(0.769253\pi\)
\(360\) 0 0
\(361\) 4.12471 7.14421i 0.217090 0.376011i
\(362\) 0 0
\(363\) −17.3179 + 7.68035i −0.908954 + 0.403114i
\(364\) 0 0
\(365\) −47.1686 + 17.1680i −2.46892 + 0.898613i
\(366\) 0 0
\(367\) −4.50518 25.5501i −0.235168 1.33371i −0.842259 0.539073i \(-0.818775\pi\)
0.607091 0.794633i \(-0.292337\pi\)
\(368\) 0 0
\(369\) 3.03525 1.60260i 0.158009 0.0834278i
\(370\) 0 0
\(371\) 4.40324 3.69475i 0.228605 0.191822i
\(372\) 0 0
\(373\) −2.08471 + 11.8230i −0.107942 + 0.612172i 0.882062 + 0.471134i \(0.156155\pi\)
−0.990004 + 0.141038i \(0.954956\pi\)
\(374\) 0 0
\(375\) 8.30731 16.9700i 0.428988 0.876328i
\(376\) 0 0
\(377\) −14.2553 −0.734184
\(378\) 0 0
\(379\) 8.30190 0.426440 0.213220 0.977004i \(-0.431605\pi\)
0.213220 + 0.977004i \(0.431605\pi\)
\(380\) 0 0
\(381\) 8.41354 + 12.5127i 0.431039 + 0.641044i
\(382\) 0 0
\(383\) 5.56949 31.5861i 0.284588 1.61398i −0.422167 0.906518i \(-0.638731\pi\)
0.706755 0.707458i \(-0.250158\pi\)
\(384\) 0 0
\(385\) −0.690227 + 0.579170i −0.0351772 + 0.0295172i
\(386\) 0 0
\(387\) 2.12372 + 6.60121i 0.107955 + 0.335558i
\(388\) 0 0
\(389\) 5.66976 + 32.1548i 0.287468 + 1.63031i 0.696332 + 0.717719i \(0.254814\pi\)
−0.408864 + 0.912595i \(0.634075\pi\)
\(390\) 0 0
\(391\) −19.7659 + 7.19419i −0.999604 + 0.363826i
\(392\) 0 0
\(393\) −28.0253 20.4237i −1.41369 1.03024i
\(394\) 0 0
\(395\) −13.9855 + 24.2236i −0.703686 + 1.21882i
\(396\) 0 0
\(397\) −12.1464 21.0382i −0.609612 1.05588i −0.991304 0.131589i \(-0.957992\pi\)
0.381692 0.924289i \(-0.375341\pi\)
\(398\) 0 0
\(399\) 5.45679 + 1.57328i 0.273181 + 0.0787623i
\(400\) 0 0
\(401\) 8.25171 + 6.92401i 0.412071 + 0.345768i 0.825137 0.564932i \(-0.191098\pi\)
−0.413066 + 0.910701i \(0.635542\pi\)
\(402\) 0 0
\(403\) −15.2474 5.54960i −0.759527 0.276445i
\(404\) 0 0
\(405\) −3.20752 32.3197i −0.159383 1.60598i
\(406\) 0 0
\(407\) 0.362184 + 0.131824i 0.0179528 + 0.00653428i
\(408\) 0 0
\(409\) 18.3715 + 15.4155i 0.908413 + 0.762249i 0.971816 0.235739i \(-0.0757511\pi\)
−0.0634037 + 0.997988i \(0.520196\pi\)
\(410\) 0 0
\(411\) −7.78433 2.24434i −0.383973 0.110705i
\(412\) 0 0
\(413\) 5.01076 + 8.67889i 0.246563 + 0.427060i
\(414\) 0 0
\(415\) 10.5952 18.3514i 0.520098 0.900836i
\(416\) 0 0
\(417\) −27.8033 20.2619i −1.36153 0.992230i
\(418\) 0 0
\(419\) 17.1977 6.25944i 0.840161 0.305793i 0.114139 0.993465i \(-0.463589\pi\)
0.726022 + 0.687671i \(0.241367\pi\)
\(420\) 0 0
\(421\) −3.39376 19.2470i −0.165402 0.938039i −0.948649 0.316329i \(-0.897550\pi\)
0.783248 0.621710i \(-0.213562\pi\)
\(422\) 0 0
\(423\) 22.5864 24.9389i 1.09819 1.21257i
\(424\) 0 0
\(425\) −13.9920 + 11.7407i −0.678713 + 0.569508i
\(426\) 0 0
\(427\) 1.10659 6.27579i 0.0535517 0.303707i
\(428\) 0 0
\(429\) −0.782466 1.16369i −0.0377778 0.0561834i
\(430\) 0 0
\(431\) 13.2168 0.636630 0.318315 0.947985i \(-0.396883\pi\)
0.318315 + 0.947985i \(0.396883\pi\)
\(432\) 0 0
\(433\) −0.870575 −0.0418371 −0.0209186 0.999781i \(-0.506659\pi\)
−0.0209186 + 0.999781i \(0.506659\pi\)
\(434\) 0 0
\(435\) 12.0817 24.6802i 0.579272 1.18333i
\(436\) 0 0
\(437\) 5.26040 29.8332i 0.251639 1.42711i
\(438\) 0 0
\(439\) 2.60614 2.18681i 0.124384 0.104371i −0.578474 0.815701i \(-0.696351\pi\)
0.702858 + 0.711330i \(0.251907\pi\)
\(440\) 0 0
\(441\) 0.113393 2.99786i 0.00539966 0.142755i
\(442\) 0 0
\(443\) 0.753468 + 4.27313i 0.0357983 + 0.203022i 0.997461 0.0712120i \(-0.0226867\pi\)
−0.961663 + 0.274234i \(0.911576\pi\)
\(444\) 0 0
\(445\) 48.3075 17.5825i 2.29000 0.833490i
\(446\) 0 0
\(447\) 16.3734 7.26148i 0.774436 0.343456i
\(448\) 0 0
\(449\) −8.08889 + 14.0104i −0.381738 + 0.661190i −0.991311 0.131540i \(-0.958008\pi\)
0.609573 + 0.792730i \(0.291341\pi\)
\(450\) 0 0
\(451\) 0.142832 + 0.247392i 0.00672570 + 0.0116493i
\(452\) 0 0
\(453\) −4.08025 16.4667i −0.191707 0.773672i
\(454\) 0 0
\(455\) 8.96396 + 7.52165i 0.420237 + 0.352620i
\(456\) 0 0
\(457\) −14.8374 5.40036i −0.694063 0.252618i −0.0291891 0.999574i \(-0.509292\pi\)
−0.664874 + 0.746956i \(0.731515\pi\)
\(458\) 0 0
\(459\) −3.70454 + 11.2349i −0.172913 + 0.524399i
\(460\) 0 0
\(461\) 13.0694 + 4.75687i 0.608703 + 0.221550i 0.627936 0.778265i \(-0.283900\pi\)
−0.0192327 + 0.999815i \(0.506122\pi\)
\(462\) 0 0
\(463\) 11.3862 + 9.55414i 0.529161 + 0.444019i 0.867812 0.496893i \(-0.165526\pi\)
−0.338651 + 0.940912i \(0.609970\pi\)
\(464\) 0 0
\(465\) 22.5306 21.6945i 1.04483 1.00606i
\(466\) 0 0
\(467\) 4.13484 + 7.16176i 0.191338 + 0.331407i 0.945694 0.325059i \(-0.105384\pi\)
−0.754356 + 0.656465i \(0.772051\pi\)
\(468\) 0 0
\(469\) 5.75570 9.96917i 0.265774 0.460334i
\(470\) 0 0
\(471\) 3.38856 31.7964i 0.156137 1.46510i
\(472\) 0 0
\(473\) −0.542325 + 0.197390i −0.0249361 + 0.00907601i
\(474\) 0 0
\(475\) −4.56788 25.9057i −0.209589 1.18864i
\(476\) 0 0
\(477\) −17.0831 + 2.35038i −0.782182 + 0.107616i
\(478\) 0 0
\(479\) 2.77319 2.32698i 0.126710 0.106323i −0.577230 0.816581i \(-0.695867\pi\)
0.703941 + 0.710259i \(0.251422\pi\)
\(480\) 0 0
\(481\) 0.869202 4.92949i 0.0396322 0.224765i
\(482\) 0 0
\(483\) −15.9653 + 1.09314i −0.726447 + 0.0497397i
\(484\) 0 0
\(485\) 0.315420 0.0143225
\(486\) 0 0
\(487\) −9.11319 −0.412958 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(488\) 0 0
\(489\) −36.2352 + 2.48102i −1.63861 + 0.112195i
\(490\) 0 0
\(491\) −3.79128 + 21.5014i −0.171098 + 0.970344i 0.771454 + 0.636285i \(0.219530\pi\)
−0.942552 + 0.334059i \(0.891581\pi\)
\(492\) 0 0
\(493\) −7.66717 + 6.43352i −0.345312 + 0.289751i
\(494\) 0 0
\(495\) 2.67786 0.368432i 0.120361 0.0165598i
\(496\) 0 0
\(497\) −0.139844 0.793097i −0.00627288 0.0355753i
\(498\) 0 0
\(499\) −15.0946 + 5.49398i −0.675727 + 0.245945i −0.657012 0.753880i \(-0.728180\pi\)
−0.0187153 + 0.999825i \(0.505958\pi\)
\(500\) 0 0
\(501\) −4.41741 + 41.4506i −0.197355 + 1.85188i
\(502\) 0 0
\(503\) −11.0312 + 19.1066i −0.491858 + 0.851922i −0.999956 0.00937667i \(-0.997015\pi\)
0.508098 + 0.861299i \(0.330349\pi\)
\(504\) 0 0
\(505\) 8.89157 + 15.4006i 0.395670 + 0.685320i
\(506\) 0 0
\(507\) 3.10119 2.98611i 0.137729 0.132618i
\(508\) 0 0
\(509\) 11.9959 + 10.0658i 0.531709 + 0.446157i 0.868691 0.495354i \(-0.164962\pi\)
−0.336982 + 0.941511i \(0.609406\pi\)
\(510\) 0 0
\(511\) 13.0707 + 4.75736i 0.578216 + 0.210453i
\(512\) 0 0
\(513\) −11.3449 12.7106i −0.500890 0.561185i
\(514\) 0 0
\(515\) −18.1931 6.62176i −0.801686 0.291790i
\(516\) 0 0
\(517\) 2.14516 + 1.80000i 0.0943438 + 0.0791639i
\(518\) 0 0
\(519\) 1.72931 + 6.97899i 0.0759084 + 0.306343i
\(520\) 0 0
\(521\) −18.4749 31.9994i −0.809399 1.40192i −0.913281 0.407330i \(-0.866460\pi\)
0.103882 0.994590i \(-0.466874\pi\)
\(522\) 0 0
\(523\) 7.80150 13.5126i 0.341136 0.590864i −0.643508 0.765439i \(-0.722522\pi\)
0.984644 + 0.174575i \(0.0558551\pi\)
\(524\) 0 0
\(525\) −12.7028 + 5.63359i −0.554395 + 0.245870i
\(526\) 0 0
\(527\) −10.7054 + 3.89643i −0.466333 + 0.169731i
\(528\) 0 0
\(529\) 10.8290 + 61.4146i 0.470828 + 2.67020i
\(530\) 0 0
\(531\) 1.13637 30.0431i 0.0493142 1.30376i
\(532\) 0 0
\(533\) 2.84195 2.38468i 0.123099 0.103292i
\(534\) 0 0
\(535\) 3.63405 20.6097i 0.157114 0.891036i
\(536\) 0 0
\(537\) 2.00877 4.10347i 0.0866847 0.177078i
\(538\) 0 0
\(539\) 0.249681 0.0107545
\(540\) 0 0
\(541\) −1.62538 −0.0698804 −0.0349402 0.999389i \(-0.511124\pi\)
−0.0349402 + 0.999389i \(0.511124\pi\)
\(542\) 0 0
\(543\) 24.1551 + 35.9236i 1.03659 + 1.54163i
\(544\) 0 0
\(545\) 1.30490 7.40048i 0.0558960 0.317002i
\(546\) 0 0
\(547\) −15.3360 + 12.8684i −0.655719 + 0.550213i −0.908800 0.417232i \(-0.863000\pi\)
0.253081 + 0.967445i \(0.418556\pi\)
\(548\) 0 0
\(549\) −12.8335 + 14.1702i −0.547719 + 0.604767i
\(550\) 0 0
\(551\) −2.50305 14.1955i −0.106633 0.604748i
\(552\) 0 0
\(553\) 7.28350 2.65098i 0.309726 0.112731i
\(554\) 0 0
\(555\) 7.79778 + 5.68271i 0.330997 + 0.241218i
\(556\) 0 0
\(557\) −19.6600 + 34.0521i −0.833021 + 1.44283i 0.0626106 + 0.998038i \(0.480057\pi\)
−0.895632 + 0.444797i \(0.853276\pi\)
\(558\) 0 0
\(559\) 3.74758 + 6.49101i 0.158506 + 0.274540i
\(560\) 0 0
\(561\) −0.946030 0.272755i −0.0399414 0.0115157i
\(562\) 0 0
\(563\) −13.7095 11.5037i −0.577788 0.484821i 0.306432 0.951893i \(-0.400865\pi\)
−0.884220 + 0.467071i \(0.845309\pi\)
\(564\) 0 0
\(565\) −19.7957 7.20506i −0.832813 0.303119i
\(566\) 0 0
\(567\) −5.07593 + 7.43202i −0.213169 + 0.312116i
\(568\) 0 0
\(569\) 17.4802 + 6.36226i 0.732807 + 0.266720i 0.681352 0.731956i \(-0.261392\pi\)
0.0514543 + 0.998675i \(0.483614\pi\)
\(570\) 0 0
\(571\) −16.4611 13.8125i −0.688874 0.578034i 0.229710 0.973259i \(-0.426222\pi\)
−0.918584 + 0.395225i \(0.870667\pi\)
\(572\) 0 0
\(573\) 18.3040 + 5.27733i 0.764662 + 0.220464i
\(574\) 0 0
\(575\) 37.0622 + 64.1935i 1.54560 + 2.67706i
\(576\) 0 0
\(577\) −3.92088 + 6.79117i −0.163228 + 0.282720i −0.936025 0.351934i \(-0.885524\pi\)
0.772796 + 0.634654i \(0.218857\pi\)
\(578\) 0 0
\(579\) −27.4420 19.9987i −1.14045 0.831116i
\(580\) 0 0
\(581\) −5.51788 + 2.00834i −0.228920 + 0.0833201i
\(582\) 0 0
\(583\) −0.249215 1.41337i −0.0103214 0.0585357i
\(584\) 0 0
\(585\) −10.7511 33.4180i −0.444504 1.38166i
\(586\) 0 0
\(587\) −21.3256 + 17.8943i −0.880203 + 0.738578i −0.966221 0.257715i \(-0.917030\pi\)
0.0860177 + 0.996294i \(0.472586\pi\)
\(588\) 0 0
\(589\) 2.84907 16.1579i 0.117394 0.665774i
\(590\) 0 0
\(591\) 2.40932 + 3.58316i 0.0991063 + 0.147392i
\(592\) 0 0
\(593\) −3.56593 −0.146435 −0.0732175 0.997316i \(-0.523327\pi\)
−0.0732175 + 0.997316i \(0.523327\pi\)
\(594\) 0 0
\(595\) 8.21582 0.336816
\(596\) 0 0
\(597\) 14.3470 29.3077i 0.587183 1.19949i
\(598\) 0 0
\(599\) 6.89443 39.1002i 0.281699 1.59759i −0.435146 0.900360i \(-0.643303\pi\)
0.716845 0.697233i \(-0.245586\pi\)
\(600\) 0 0
\(601\) −15.4289 + 12.9464i −0.629359 + 0.528095i −0.900730 0.434380i \(-0.856968\pi\)
0.271370 + 0.962475i \(0.412523\pi\)
\(602\) 0 0
\(603\) −30.5388 + 16.1243i −1.24364 + 0.656634i
\(604\) 0 0
\(605\) −6.85406 38.8713i −0.278657 1.58034i
\(606\) 0 0
\(607\) 24.6299 8.96455i 0.999697 0.363860i 0.210229 0.977652i \(-0.432579\pi\)
0.789468 + 0.613792i \(0.210357\pi\)
\(608\) 0 0
\(609\) −6.96071 + 3.08702i −0.282062 + 0.125092i
\(610\) 0 0
\(611\) 18.1837 31.4951i 0.735634 1.27416i
\(612\) 0 0
\(613\) 15.8806 + 27.5061i 0.641413 + 1.11096i 0.985117 + 0.171882i \(0.0549849\pi\)
−0.343704 + 0.939078i \(0.611682\pi\)
\(614\) 0 0
\(615\) 1.71999 + 6.94136i 0.0693566 + 0.279903i
\(616\) 0 0
\(617\) 1.16959 + 0.981406i 0.0470861 + 0.0395099i 0.666027 0.745928i \(-0.267994\pi\)
−0.618941 + 0.785438i \(0.712438\pi\)
\(618\) 0 0
\(619\) 19.9875 + 7.27486i 0.803366 + 0.292401i 0.710881 0.703313i \(-0.248297\pi\)
0.0924855 + 0.995714i \(0.470519\pi\)
\(620\) 0 0
\(621\) 42.2902 + 22.7226i 1.69705 + 0.911826i
\(622\) 0 0
\(623\) −13.3863 4.87223i −0.536312 0.195202i
\(624\) 0 0
\(625\) −0.573125 0.480909i −0.0229250 0.0192364i
\(626\) 0 0
\(627\) 1.02142 0.983513i 0.0407915 0.0392777i
\(628\) 0 0
\(629\) −1.75722 3.04359i −0.0700649 0.121356i
\(630\) 0 0
\(631\) 14.0390 24.3162i 0.558883 0.968014i −0.438707 0.898630i \(-0.644563\pi\)
0.997590 0.0693835i \(-0.0221032\pi\)
\(632\) 0 0
\(633\) −1.87964 + 17.6375i −0.0747090 + 0.701029i
\(634\) 0 0
\(635\) −29.5209 + 10.7447i −1.17150 + 0.426392i
\(636\) 0 0
\(637\) −0.563071 3.19333i −0.0223097 0.126524i
\(638\) 0 0
\(639\) −0.911540 + 2.23744i −0.0360600 + 0.0885117i
\(640\) 0 0
\(641\) 2.46985 2.07245i 0.0975534 0.0818570i −0.592707 0.805418i \(-0.701941\pi\)
0.690260 + 0.723561i \(0.257496\pi\)
\(642\) 0 0
\(643\) −4.12078 + 23.3701i −0.162508 + 0.921627i 0.789089 + 0.614279i \(0.210553\pi\)
−0.951597 + 0.307349i \(0.900558\pi\)
\(644\) 0 0
\(645\) −14.4141 + 0.986929i −0.567553 + 0.0388603i
\(646\) 0 0
\(647\) −23.2970 −0.915900 −0.457950 0.888978i \(-0.651416\pi\)
−0.457950 + 0.888978i \(0.651416\pi\)
\(648\) 0 0
\(649\) 2.50218 0.0982192
\(650\) 0 0
\(651\) −8.64694 + 0.592054i −0.338900 + 0.0232044i
\(652\) 0 0
\(653\) −5.74561 + 32.5850i −0.224843 + 1.27515i 0.638142 + 0.769919i \(0.279703\pi\)
−0.862985 + 0.505230i \(0.831408\pi\)
\(654\) 0 0
\(655\) 55.3474 46.4420i 2.16260 1.81464i
\(656\) 0 0
\(657\) −25.5953 32.9571i −0.998569 1.28578i
\(658\) 0 0
\(659\) 4.83102 + 27.3981i 0.188190 + 1.06728i 0.921788 + 0.387694i \(0.126728\pi\)
−0.733599 + 0.679583i \(0.762161\pi\)
\(660\) 0 0
\(661\) 18.0476 6.56879i 0.701971 0.255496i 0.0337185 0.999431i \(-0.489265\pi\)
0.668252 + 0.743935i \(0.267043\pi\)
\(662\) 0 0
\(663\) −1.35499 + 12.7145i −0.0526235 + 0.493790i
\(664\) 0 0
\(665\) −5.91615 + 10.2471i −0.229418 + 0.397364i
\(666\) 0 0
\(667\) 20.3088 + 35.1759i 0.786361 + 1.36202i
\(668\) 0 0
\(669\) −26.7329 + 25.7408i −1.03355 + 0.995198i
\(670\) 0 0
\(671\) −1.21887 1.02275i −0.0470538 0.0394829i
\(672\) 0 0
\(673\) 22.5092 + 8.19269i 0.867667 + 0.315805i 0.737222 0.675651i \(-0.236137\pi\)
0.130445 + 0.991456i \(0.458359\pi\)
\(674\) 0 0
\(675\) 41.2566 + 5.98136i 1.58797 + 0.230223i
\(676\) 0 0
\(677\) −13.0970 4.76692i −0.503359 0.183208i 0.0778452 0.996965i \(-0.475196\pi\)
−0.581204 + 0.813758i \(0.697418\pi\)
\(678\) 0 0
\(679\) −0.0669560 0.0561828i −0.00256954 0.00215610i
\(680\) 0 0
\(681\) 4.98244 + 20.1077i 0.190928 + 0.770527i
\(682\) 0 0
\(683\) 2.66819 + 4.62144i 0.102095 + 0.176835i 0.912548 0.408970i \(-0.134112\pi\)
−0.810452 + 0.585805i \(0.800779\pi\)
\(684\) 0 0
\(685\) 8.43963 14.6179i 0.322462 0.558520i
\(686\) 0 0
\(687\) −10.6035 + 4.70258i −0.404550 + 0.179415i
\(688\) 0 0
\(689\) −17.5145 + 6.37474i −0.667248 + 0.242858i
\(690\) 0 0
\(691\) 3.28113 + 18.6082i 0.124820 + 0.707891i 0.981414 + 0.191900i \(0.0614650\pi\)
−0.856594 + 0.515991i \(0.827424\pi\)
\(692\) 0 0
\(693\) −0.634070 0.398773i −0.0240863 0.0151481i
\(694\) 0 0
\(695\) 54.9089 46.0740i 2.08281 1.74769i
\(696\) 0 0
\(697\) 0.452313 2.56519i 0.0171326 0.0971636i
\(698\) 0 0
\(699\) −14.3351 + 29.2834i −0.542202 + 1.10760i
\(700\) 0 0
\(701\) 11.5259 0.435328 0.217664 0.976024i \(-0.430156\pi\)
0.217664 + 0.976024i \(0.430156\pi\)
\(702\) 0 0
\(703\) 5.06144 0.190896
\(704\) 0 0
\(705\) 39.1165 + 58.1743i 1.47321 + 2.19097i
\(706\) 0 0
\(707\) 0.855708 4.85296i 0.0321822 0.182514i
\(708\) 0 0
\(709\) −21.0898 + 17.6965i −0.792045 + 0.664605i −0.946251 0.323434i \(-0.895163\pi\)
0.154205 + 0.988039i \(0.450718\pi\)
\(710\) 0 0
\(711\) −22.7306 4.90047i −0.852463 0.183782i
\(712\) 0 0
\(713\) 8.02823 + 45.5304i 0.300660 + 1.70513i
\(714\) 0 0
\(715\) 2.74547 0.999270i 0.102675 0.0373706i
\(716\) 0 0
\(717\) −20.6056 15.0165i −0.769529 0.560802i
\(718\) 0 0
\(719\) −3.06437 + 5.30764i −0.114282 + 0.197942i −0.917492 0.397753i \(-0.869790\pi\)
0.803211 + 0.595695i \(0.203123\pi\)
\(720\) 0 0
\(721\) 2.68249 + 4.64622i 0.0999013 + 0.173034i
\(722\) 0 0
\(723\) −33.9056 9.77551i −1.26096 0.363555i
\(724\) 0 0
\(725\) 27.0187 + 22.6714i 1.00345 + 0.841995i
\(726\) 0 0
\(727\) 20.6802 + 7.52699i 0.766988 + 0.279161i 0.695736 0.718298i \(-0.255078\pi\)
0.0712519 + 0.997458i \(0.477301\pi\)
\(728\) 0 0
\(729\) 24.7603 10.7669i 0.917050 0.398773i
\(730\) 0 0
\(731\) 4.94507 + 1.79986i 0.182900 + 0.0665702i
\(732\) 0 0
\(733\) −0.366420 0.307463i −0.0135340 0.0113564i 0.635996 0.771693i \(-0.280590\pi\)
−0.649530 + 0.760336i \(0.725034\pi\)
\(734\) 0 0
\(735\) 6.00585 + 1.73158i 0.221529 + 0.0638702i
\(736\) 0 0
\(737\) −1.43709 2.48911i −0.0529359 0.0916876i
\(738\) 0 0
\(739\) 9.56120 16.5605i 0.351715 0.609187i −0.634835 0.772647i \(-0.718932\pi\)
0.986550 + 0.163460i \(0.0522654\pi\)
\(740\) 0 0
\(741\) −14.8823 10.8456i −0.546713 0.398423i
\(742\) 0 0
\(743\) 4.59805 1.67355i 0.168686 0.0613967i −0.256297 0.966598i \(-0.582502\pi\)
0.424983 + 0.905201i \(0.360280\pi\)
\(744\) 0 0
\(745\) 6.48025 + 36.7513i 0.237418 + 1.34646i
\(746\) 0 0
\(747\) 17.2204 + 3.71253i 0.630060 + 0.135834i
\(748\) 0 0
\(749\) −4.44244 + 3.72765i −0.162323 + 0.136205i
\(750\) 0 0
\(751\) −5.13742 + 29.1357i −0.187467 + 1.06318i 0.735277 + 0.677766i \(0.237052\pi\)
−0.922744 + 0.385412i \(0.874059\pi\)
\(752\) 0 0
\(753\) 19.0014 + 28.2590i 0.692448 + 1.02981i
\(754\) 0 0
\(755\) 35.3458 1.28636
\(756\) 0 0
\(757\) 49.0230 1.78177 0.890886 0.454226i \(-0.150084\pi\)
0.890886 + 0.454226i \(0.150084\pi\)
\(758\) 0 0
\(759\) −1.75675 + 3.58865i −0.0637658 + 0.130260i
\(760\) 0 0
\(761\) 3.77770 21.4244i 0.136942 0.776634i −0.836546 0.547896i \(-0.815429\pi\)
0.973488 0.228738i \(-0.0734599\pi\)
\(762\) 0 0
\(763\) −1.59518 + 1.33851i −0.0577494 + 0.0484575i
\(764\) 0 0
\(765\) −20.8643 13.1217i −0.754350 0.474417i
\(766\) 0 0
\(767\) −5.64282 32.0020i −0.203751 1.15553i
\(768\) 0 0
\(769\) −7.16325 + 2.60721i −0.258313 + 0.0940184i −0.467931 0.883765i \(-0.655000\pi\)
0.209617 + 0.977783i \(0.432778\pi\)
\(770\) 0 0
\(771\) 6.73057 2.98496i 0.242396 0.107501i
\(772\) 0 0
\(773\) 9.75263 16.8921i 0.350778 0.607565i −0.635608 0.772012i \(-0.719251\pi\)
0.986386 + 0.164447i \(0.0525839\pi\)
\(774\) 0 0
\(775\) 20.0732 + 34.7677i 0.721049 + 1.24889i
\(776\) 0 0
\(777\) −0.643073 2.59525i −0.0230701 0.0931041i
\(778\) 0 0
\(779\) 2.87369 + 2.41131i 0.102961 + 0.0863943i
\(780\) 0 0
\(781\) −0.188950 0.0687720i −0.00676115 0.00246086i
\(782\) 0 0
\(783\) 22.6072 + 3.27759i 0.807917 + 0.117131i
\(784\) 0 0
\(785\) 62.6050 + 22.7863i 2.23447 + 0.813280i
\(786\) 0 0
\(787\) −3.40233 2.85489i −0.121280 0.101766i 0.580131 0.814523i \(-0.303002\pi\)
−0.701410 + 0.712758i \(0.747446\pi\)
\(788\) 0 0
\(789\) 25.6735 24.7208i 0.914002 0.880084i
\(790\) 0 0
\(791\) 2.91879 + 5.05549i 0.103780 + 0.179753i
\(792\) 0 0
\(793\) −10.3319 + 17.8954i −0.366896 + 0.635483i
\(794\) 0 0
\(795\) 3.80729 35.7256i 0.135031 1.26706i
\(796\) 0 0
\(797\) −14.5510 + 5.29613i −0.515423 + 0.187598i −0.586618 0.809864i \(-0.699541\pi\)
0.0711952 + 0.997462i \(0.477319\pi\)
\(798\) 0 0
\(799\) −4.43392 25.1460i −0.156861 0.889602i
\(800\) 0 0
\(801\) 26.2133 + 33.7528i 0.926203 + 1.19260i
\(802\) 0 0
\(803\) 2.66044 2.23238i 0.0938849 0.0787788i
\(804\) 0 0
\(805\) 5.78969 32.8350i 0.204060 1.15728i
\(806\) 0 0
\(807\) −4.78443 + 0.327589i −0.168420 + 0.0115317i
\(808\) 0 0
\(809\) −48.0670 −1.68994 −0.844972 0.534810i \(-0.820383\pi\)
−0.844972 + 0.534810i \(0.820383\pi\)
\(810\) 0 0
\(811\) −9.75239 −0.342453 −0.171226 0.985232i \(-0.554773\pi\)
−0.171226 + 0.985232i \(0.554773\pi\)
\(812\) 0 0
\(813\) −7.98022 + 0.546404i −0.279879 + 0.0191632i
\(814\) 0 0
\(815\) 13.1404 74.5229i 0.460288 2.61042i
\(816\) 0 0
\(817\) −5.80575 + 4.87161i −0.203118 + 0.170436i
\(818\) 0 0
\(819\) −3.67023 + 9.00884i −0.128248 + 0.314794i
\(820\) 0 0
\(821\) 3.63063 + 20.5903i 0.126710 + 0.718607i 0.980278 + 0.197625i \(0.0633229\pi\)
−0.853568 + 0.520982i \(0.825566\pi\)
\(822\) 0 0
\(823\) −18.4122 + 6.70149i −0.641809 + 0.233599i −0.642363 0.766400i \(-0.722046\pi\)
0.000554035 1.00000i \(0.499824\pi\)
\(824\) 0 0
\(825\) −0.367671 + 3.45002i −0.0128007 + 0.120114i
\(826\) 0 0
\(827\) 13.8685 24.0210i 0.482256 0.835292i −0.517536 0.855661i \(-0.673151\pi\)
0.999793 + 0.0203691i \(0.00648412\pi\)
\(828\) 0 0
\(829\) 7.59903 + 13.1619i 0.263925 + 0.457132i 0.967281 0.253706i \(-0.0816495\pi\)
−0.703356 + 0.710837i \(0.748316\pi\)
\(830\) 0 0
\(831\) −35.9706 + 34.6358i −1.24781 + 1.20150i
\(832\) 0 0
\(833\) −1.74402 1.46341i −0.0604268 0.0507041i
\(834\) 0 0
\(835\) −81.6133 29.7048i −2.82435 1.02798i
\(836\) 0 0
\(837\) 22.9047 + 12.3067i 0.791702 + 0.425383i
\(838\) 0 0
\(839\) 30.9531 + 11.2660i 1.06862 + 0.388946i 0.815661 0.578531i \(-0.196374\pi\)
0.252960 + 0.967477i \(0.418596\pi\)
\(840\) 0 0
\(841\) −7.40992 6.21766i −0.255514 0.214402i
\(842\) 0 0
\(843\) 13.5456 + 54.6658i 0.466534 + 1.88279i
\(844\) 0 0
\(845\) 4.48487 + 7.76802i 0.154284 + 0.267228i
\(846\) 0 0
\(847\) −5.46883 + 9.47229i −0.187911 + 0.325472i
\(848\) 0 0
\(849\) 31.1081 13.7962i 1.06763 0.473484i
\(850\) 0 0
\(851\) −13.4022 + 4.87800i −0.459421 + 0.167216i
\(852\) 0 0
\(853\) −6.40622 36.3315i −0.219345 1.24397i −0.873206 0.487352i \(-0.837963\pi\)
0.653861 0.756615i \(-0.273148\pi\)
\(854\) 0 0
\(855\) 31.3901 16.5738i 1.07352 0.566812i
\(856\) 0 0
\(857\) −30.1224 + 25.2757i −1.02896 + 0.863400i −0.990727 0.135869i \(-0.956617\pi\)
−0.0382333 + 0.999269i \(0.512173\pi\)
\(858\) 0 0
\(859\) 3.87616 21.9828i 0.132253 0.750044i −0.844480 0.535586i \(-0.820091\pi\)
0.976733 0.214457i \(-0.0687983\pi\)
\(860\) 0 0
\(861\) 0.871288 1.77985i 0.0296934 0.0606571i
\(862\) 0 0
\(863\) −2.07889 −0.0707662 −0.0353831 0.999374i \(-0.511265\pi\)
−0.0353831 + 0.999374i \(0.511265\pi\)
\(864\) 0 0
\(865\) −14.9804 −0.509350
\(866\) 0 0
\(867\) −11.4206 16.9848i −0.387863 0.576833i
\(868\) 0 0
\(869\) 0.336055 1.90586i 0.0113999 0.0646519i
\(870\) 0 0
\(871\) −28.5940 + 23.9932i −0.968871 + 0.812980i
\(872\) 0 0
\(873\) 0.0803052 + 0.249615i 0.00271792 + 0.00844819i
\(874\) 0 0
\(875\) −1.89426 10.7429i −0.0640376 0.363176i
\(876\) 0 0
\(877\) −31.4991 + 11.4647i −1.06365 + 0.387136i −0.813798 0.581148i \(-0.802604\pi\)
−0.249851 + 0.968284i \(0.580382\pi\)
\(878\) 0 0
\(879\) −36.3657 26.5019i −1.22658 0.893886i
\(880\) 0 0
\(881\) −13.5648 + 23.4949i −0.457009 + 0.791564i −0.998801 0.0489491i \(-0.984413\pi\)
0.541792 + 0.840513i \(0.317746\pi\)
\(882\) 0 0
\(883\) 7.22889 + 12.5208i 0.243272 + 0.421359i 0.961644 0.274300i \(-0.0884460\pi\)
−0.718373 + 0.695659i \(0.755113\pi\)
\(884\) 0 0
\(885\) 60.1877 + 17.3530i 2.02319 + 0.583316i
\(886\) 0 0
\(887\) −31.8733 26.7449i −1.07020 0.898005i −0.0751301 0.997174i \(-0.523937\pi\)
−0.995071 + 0.0991687i \(0.968382\pi\)
\(888\) 0 0
\(889\) 8.18045 + 2.97744i 0.274364 + 0.0998602i
\(890\) 0 0
\(891\) 0.973345 + 2.02538i 0.0326083 + 0.0678529i
\(892\) 0 0
\(893\) 34.5558 + 12.5773i 1.15637 + 0.420883i
\(894\) 0 0
\(895\) 7.29197 + 6.11869i 0.243744 + 0.204525i
\(896\) 0 0
\(897\) 49.8593 + 14.3752i 1.66475 + 0.479974i
\(898\) 0 0
\(899\) 10.9994 + 19.0516i 0.366851 + 0.635405i
\(900\) 0 0
\(901\) −6.54314 + 11.3331i −0.217984 + 0.377559i
\(902\) 0 0
\(903\) 3.23555 + 2.35794i 0.107672 + 0.0784674i
\(904\) 0 0
\(905\) −84.7540 + 30.8479i −2.81732 + 1.02542i
\(906\) 0 0
\(907\) −2.52575 14.3242i −0.0838661 0.475628i −0.997596 0.0693051i \(-0.977922\pi\)
0.913729 0.406323i \(-0.133189\pi\)
\(908\) 0 0
\(909\) −9.92390 + 10.9575i −0.329155 + 0.363438i
\(910\) 0 0
\(911\) −3.97729 + 3.33734i −0.131774 + 0.110571i −0.706292 0.707920i \(-0.749634\pi\)
0.574519 + 0.818491i \(0.305189\pi\)
\(912\) 0 0
\(913\) −0.254590 + 1.44385i −0.00842571 + 0.0477846i
\(914\) 0 0
\(915\) −22.2258 33.0544i −0.734762 1.09274i
\(916\) 0 0
\(917\) −20.0212 −0.661158
\(918\) 0 0
\(919\) −27.5329 −0.908225 −0.454112 0.890944i \(-0.650044\pi\)
−0.454112 + 0.890944i \(0.650044\pi\)
\(920\) 0 0
\(921\) −21.6087 + 44.1418i −0.712030 + 1.45452i
\(922\) 0 0
\(923\) −0.453459 + 2.57169i −0.0149258 + 0.0846483i
\(924\) 0 0
\(925\) −9.48725 + 7.96075i −0.311939 + 0.261748i
\(926\) 0 0
\(927\) 0.608352 16.0835i 0.0199809 0.528250i
\(928\) 0 0
\(929\) −0.841372 4.77166i −0.0276045 0.156553i 0.967890 0.251375i \(-0.0808828\pi\)
−0.995494 + 0.0948223i \(0.969772\pi\)
\(930\) 0 0
\(931\) 3.08107 1.12142i 0.100978 0.0367530i
\(932\) 0 0
\(933\) −31.6569 + 14.0396i −1.03640 + 0.459635i
\(934\) 0 0
\(935\) 1.02567 1.77651i 0.0335429 0.0580980i
\(936\) 0 0
\(937\) −16.2565 28.1571i −0.531078 0.919854i −0.999342 0.0362654i \(-0.988454\pi\)
0.468264 0.883588i \(-0.344879\pi\)
\(938\) 0 0
\(939\) −10.9035 44.0032i −0.355821 1.43599i
\(940\) 0 0
\(941\) 30.3804 + 25.4922i 0.990372 + 0.831021i 0.985622 0.168968i \(-0.0540434\pi\)
0.00475044 + 0.999989i \(0.498488\pi\)
\(942\) 0 0
\(943\) −9.93318 3.61538i −0.323469 0.117733i
\(944\) 0 0
\(945\) −12.4864 13.9895i −0.406184 0.455078i
\(946\) 0 0
\(947\) −0.388535 0.141415i −0.0126257 0.00459537i 0.335700 0.941969i \(-0.391027\pi\)
−0.348325 + 0.937374i \(0.613250\pi\)
\(948\) 0 0
\(949\) −34.5510 28.9918i −1.12157 0.941113i
\(950\) 0 0
\(951\) 10.2204 9.84111i 0.331419 0.319120i
\(952\) 0 0
\(953\) −23.3738 40.4845i −0.757151 1.31142i −0.944298 0.329091i \(-0.893258\pi\)
0.187148 0.982332i \(-0.440076\pi\)
\(954\) 0 0
\(955\) −19.8449 + 34.3723i −0.642165 + 1.11226i
\(956\) 0 0
\(957\) −0.201471 + 1.89050i −0.00651264 + 0.0611111i
\(958\) 0 0
\(959\) −4.39528 + 1.59975i −0.141931 + 0.0516586i
\(960\) 0 0
\(961\) −1.03494 5.86944i −0.0333852 0.189337i
\(962\) 0 0
\(963\) 17.2352 2.37130i 0.555397 0.0764141i
\(964\) 0 0
\(965\) 54.1955 45.4754i 1.74461 1.46391i
\(966\) 0 0
\(967\) −0.960175 + 5.44542i −0.0308771 + 0.175113i −0.996346 0.0854040i \(-0.972782\pi\)
0.965469 + 0.260517i \(0.0838930\pi\)
\(968\) 0 0
\(969\) −12.8991 + 0.883199i −0.414379 + 0.0283724i
\(970\) 0 0
\(971\) −20.8628 −0.669520 −0.334760 0.942303i \(-0.608655\pi\)
−0.334760 + 0.942303i \(0.608655\pi\)
\(972\) 0 0
\(973\) −19.8626 −0.636765
\(974\) 0 0
\(975\) 44.9538 3.07798i 1.43967 0.0985742i
\(976\) 0 0
\(977\) −5.38149 + 30.5200i −0.172169 + 0.976420i 0.769192 + 0.639018i \(0.220659\pi\)
−0.941361 + 0.337402i \(0.890452\pi\)
\(978\) 0 0
\(979\) −2.72468 + 2.28628i −0.0870811 + 0.0730697i
\(980\) 0 0
\(981\) 6.18877 0.851481i 0.197592 0.0271857i
\(982\) 0 0
\(983\) −1.85075 10.4961i −0.0590297 0.334774i 0.940963 0.338509i \(-0.109922\pi\)
−0.999993 + 0.00373436i \(0.998811\pi\)
\(984\) 0 0
\(985\) −8.45370 + 3.07689i −0.269357 + 0.0980380i
\(986\) 0 0
\(987\) 2.05857 19.3165i 0.0655249 0.614850i
\(988\) 0 0
\(989\) 10.6780 18.4949i 0.339541 0.588103i
\(990\) 0 0
\(991\) −3.92160 6.79242i −0.124574 0.215768i 0.796992 0.603989i \(-0.206423\pi\)
−0.921566 + 0.388221i \(0.873090\pi\)
\(992\) 0 0
\(993\) 41.7739 40.2237i 1.32566 1.27646i
\(994\) 0 0
\(995\) 52.0806 + 43.7008i 1.65107 + 1.38541i
\(996\) 0 0
\(997\) 33.4749 + 12.1839i 1.06016 + 0.385867i 0.812489 0.582977i \(-0.198112\pi\)
0.247672 + 0.968844i \(0.420334\pi\)
\(998\) 0 0
\(999\) −2.51185 + 7.61777i −0.0794714 + 0.241015i
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 756.2.bo.b.85.1 54
27.7 even 9 inner 756.2.bo.b.169.1 yes 54
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.bo.b.85.1 54 1.1 even 1 trivial
756.2.bo.b.169.1 yes 54 27.7 even 9 inner