Properties

Label 256.2.i.a.81.1
Level $256$
Weight $2$
Character 256.81
Analytic conductor $2.044$
Analytic rank $0$
Dimension $56$
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] = [256,2,Mod(17,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.i (of order \(16\), degree \(8\), not 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: \(2.04417029174\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{16})\)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 81.1
Character \(\chi\) \(=\) 256.81
Dual form 256.2.i.a.177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.553854 + 2.78441i) q^{3} +(2.59756 + 1.73564i) q^{5} +(1.96508 - 0.813965i) q^{7} +(-4.67456 - 1.93627i) q^{9} +O(q^{10})\) \(q+(-0.553854 + 2.78441i) q^{3} +(2.59756 + 1.73564i) q^{5} +(1.96508 - 0.813965i) q^{7} +(-4.67456 - 1.93627i) q^{9} +(2.02382 - 0.402563i) q^{11} +(-1.99644 + 1.33398i) q^{13} +(-6.27140 + 6.27140i) q^{15} +(-2.31120 - 2.31120i) q^{17} +(-2.37703 - 3.55748i) q^{19} +(1.17804 + 5.92242i) q^{21} +(-0.993213 + 2.39783i) q^{23} +(1.82149 + 4.39746i) q^{25} +(3.24866 - 4.86196i) q^{27} +(2.34079 + 0.465613i) q^{29} -1.81094i q^{31} +5.85811i q^{33} +(6.51718 + 1.29635i) q^{35} +(1.40742 - 2.10635i) q^{37} +(-2.60861 - 6.29775i) q^{39} +(4.66056 - 11.2516i) q^{41} +(-0.452227 - 2.27350i) q^{43} +(-8.78181 - 13.1429i) q^{45} +(2.27298 + 2.27298i) q^{47} +(-1.75073 + 1.75073i) q^{49} +(7.71539 - 5.15526i) q^{51} +(7.98970 - 1.58925i) q^{53} +(5.95570 + 2.46693i) q^{55} +(11.2220 - 4.64831i) q^{57} +(-7.08029 - 4.73090i) q^{59} +(-1.95998 + 9.85346i) q^{61} -10.7620 q^{63} -7.50119 q^{65} +(-2.88943 + 14.5262i) q^{67} +(-6.12645 - 4.09356i) q^{69} +(6.41451 - 2.65698i) q^{71} +(-3.36337 - 1.39315i) q^{73} +(-13.2532 + 2.63622i) q^{75} +(3.64930 - 2.43839i) q^{77} +(10.9490 - 10.9490i) q^{79} +(1.00516 + 1.00516i) q^{81} +(3.72793 + 5.57925i) q^{83} +(-1.99208 - 10.0149i) q^{85} +(-2.59292 + 6.25985i) q^{87} +(-3.10595 - 7.49842i) q^{89} +(-2.83736 + 4.24642i) q^{91} +(5.04240 + 1.00299i) q^{93} -13.3664i q^{95} +8.49286i q^{97} +(-10.2399 - 2.03685i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 8 q^{3} - 8 q^{5} + 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q + 8 q^{3} - 8 q^{5} + 8 q^{7} - 8 q^{9} + 8 q^{11} - 8 q^{13} + 8 q^{15} - 8 q^{17} + 8 q^{19} - 8 q^{21} + 8 q^{23} - 8 q^{25} + 8 q^{27} - 8 q^{29} + 8 q^{35} - 8 q^{37} + 8 q^{39} - 8 q^{41} + 8 q^{43} - 8 q^{45} + 8 q^{47} - 8 q^{49} - 24 q^{51} - 8 q^{53} - 56 q^{55} - 8 q^{57} - 56 q^{59} - 8 q^{61} - 64 q^{63} - 16 q^{65} - 72 q^{67} - 8 q^{69} - 56 q^{71} - 8 q^{73} - 56 q^{75} - 8 q^{77} - 24 q^{79} - 8 q^{81} + 8 q^{83} - 8 q^{85} + 8 q^{87} - 8 q^{89} + 8 q^{91} + 16 q^{93} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(e\left(\frac{3}{16}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.553854 + 2.78441i −0.319768 + 1.60758i 0.402131 + 0.915582i \(0.368270\pi\)
−0.721898 + 0.691999i \(0.756730\pi\)
\(4\) 0 0
\(5\) 2.59756 + 1.73564i 1.16167 + 0.776200i 0.978371 0.206858i \(-0.0663237\pi\)
0.183295 + 0.983058i \(0.441324\pi\)
\(6\) 0 0
\(7\) 1.96508 0.813965i 0.742732 0.307650i 0.0209598 0.999780i \(-0.493328\pi\)
0.721772 + 0.692131i \(0.243328\pi\)
\(8\) 0 0
\(9\) −4.67456 1.93627i −1.55819 0.645422i
\(10\) 0 0
\(11\) 2.02382 0.402563i 0.610205 0.121377i 0.119692 0.992811i \(-0.461809\pi\)
0.490513 + 0.871434i \(0.336809\pi\)
\(12\) 0 0
\(13\) −1.99644 + 1.33398i −0.553713 + 0.369979i −0.800724 0.599033i \(-0.795552\pi\)
0.247011 + 0.969013i \(0.420552\pi\)
\(14\) 0 0
\(15\) −6.27140 + 6.27140i −1.61927 + 1.61927i
\(16\) 0 0
\(17\) −2.31120 2.31120i −0.560547 0.560547i 0.368916 0.929463i \(-0.379729\pi\)
−0.929463 + 0.368916i \(0.879729\pi\)
\(18\) 0 0
\(19\) −2.37703 3.55748i −0.545328 0.816142i 0.451780 0.892129i \(-0.350789\pi\)
−0.997109 + 0.0759875i \(0.975789\pi\)
\(20\) 0 0
\(21\) 1.17804 + 5.92242i 0.257070 + 1.29238i
\(22\) 0 0
\(23\) −0.993213 + 2.39783i −0.207099 + 0.499982i −0.992964 0.118417i \(-0.962218\pi\)
0.785865 + 0.618398i \(0.212218\pi\)
\(24\) 0 0
\(25\) 1.82149 + 4.39746i 0.364297 + 0.879491i
\(26\) 0 0
\(27\) 3.24866 4.86196i 0.625205 0.935685i
\(28\) 0 0
\(29\) 2.34079 + 0.465613i 0.434674 + 0.0864621i 0.407577 0.913171i \(-0.366374\pi\)
0.0270970 + 0.999633i \(0.491374\pi\)
\(30\) 0 0
\(31\) 1.81094i 0.325254i −0.986688 0.162627i \(-0.948003\pi\)
0.986688 0.162627i \(-0.0519967\pi\)
\(32\) 0 0
\(33\) 5.85811i 1.01977i
\(34\) 0 0
\(35\) 6.51718 + 1.29635i 1.10160 + 0.219123i
\(36\) 0 0
\(37\) 1.40742 2.10635i 0.231378 0.346282i −0.697554 0.716533i \(-0.745728\pi\)
0.928932 + 0.370250i \(0.120728\pi\)
\(38\) 0 0
\(39\) −2.60861 6.29775i −0.417712 1.00845i
\(40\) 0 0
\(41\) 4.66056 11.2516i 0.727858 1.75720i 0.0782501 0.996934i \(-0.475067\pi\)
0.649607 0.760270i \(-0.274933\pi\)
\(42\) 0 0
\(43\) −0.452227 2.27350i −0.0689639 0.346705i 0.930862 0.365371i \(-0.119058\pi\)
−0.999826 + 0.0186663i \(0.994058\pi\)
\(44\) 0 0
\(45\) −8.78181 13.1429i −1.30912 1.95923i
\(46\) 0 0
\(47\) 2.27298 + 2.27298i 0.331549 + 0.331549i 0.853174 0.521626i \(-0.174674\pi\)
−0.521626 + 0.853174i \(0.674674\pi\)
\(48\) 0 0
\(49\) −1.75073 + 1.75073i −0.250104 + 0.250104i
\(50\) 0 0
\(51\) 7.71539 5.15526i 1.08037 0.721881i
\(52\) 0 0
\(53\) 7.98970 1.58925i 1.09747 0.218300i 0.387036 0.922065i \(-0.373499\pi\)
0.710433 + 0.703764i \(0.248499\pi\)
\(54\) 0 0
\(55\) 5.95570 + 2.46693i 0.803067 + 0.332641i
\(56\) 0 0
\(57\) 11.2220 4.64831i 1.48639 0.615684i
\(58\) 0 0
\(59\) −7.08029 4.73090i −0.921775 0.615911i 0.00152111 0.999999i \(-0.499516\pi\)
−0.923296 + 0.384088i \(0.874516\pi\)
\(60\) 0 0
\(61\) −1.95998 + 9.85346i −0.250949 + 1.26161i 0.625543 + 0.780190i \(0.284878\pi\)
−0.876492 + 0.481417i \(0.840122\pi\)
\(62\) 0 0
\(63\) −10.7620 −1.35588
\(64\) 0 0
\(65\) −7.50119 −0.930408
\(66\) 0 0
\(67\) −2.88943 + 14.5262i −0.353000 + 1.77465i 0.241342 + 0.970440i \(0.422413\pi\)
−0.594342 + 0.804212i \(0.702587\pi\)
\(68\) 0 0
\(69\) −6.12645 4.09356i −0.737537 0.492807i
\(70\) 0 0
\(71\) 6.41451 2.65698i 0.761263 0.315325i 0.0319351 0.999490i \(-0.489833\pi\)
0.729328 + 0.684165i \(0.239833\pi\)
\(72\) 0 0
\(73\) −3.36337 1.39315i −0.393653 0.163056i 0.177072 0.984198i \(-0.443338\pi\)
−0.570724 + 0.821142i \(0.693338\pi\)
\(74\) 0 0
\(75\) −13.2532 + 2.63622i −1.53034 + 0.304404i
\(76\) 0 0
\(77\) 3.64930 2.43839i 0.415877 0.277880i
\(78\) 0 0
\(79\) 10.9490 10.9490i 1.23186 1.23186i 0.268617 0.963247i \(-0.413433\pi\)
0.963247 0.268617i \(-0.0865667\pi\)
\(80\) 0 0
\(81\) 1.00516 + 1.00516i 0.111685 + 0.111685i
\(82\) 0 0
\(83\) 3.72793 + 5.57925i 0.409194 + 0.612402i 0.977632 0.210324i \(-0.0674520\pi\)
−0.568438 + 0.822726i \(0.692452\pi\)
\(84\) 0 0
\(85\) −1.99208 10.0149i −0.216072 1.08627i
\(86\) 0 0
\(87\) −2.59292 + 6.25985i −0.277990 + 0.671126i
\(88\) 0 0
\(89\) −3.10595 7.49842i −0.329230 0.794831i −0.998650 0.0519471i \(-0.983457\pi\)
0.669420 0.742884i \(-0.266543\pi\)
\(90\) 0 0
\(91\) −2.83736 + 4.24642i −0.297437 + 0.445145i
\(92\) 0 0
\(93\) 5.04240 + 1.00299i 0.522872 + 0.104006i
\(94\) 0 0
\(95\) 13.3664i 1.37137i
\(96\) 0 0
\(97\) 8.49286i 0.862319i 0.902276 + 0.431160i \(0.141895\pi\)
−0.902276 + 0.431160i \(0.858105\pi\)
\(98\) 0 0
\(99\) −10.2399 2.03685i −1.02915 0.204711i
\(100\) 0 0
\(101\) −8.24152 + 12.3343i −0.820062 + 1.22731i 0.151013 + 0.988532i \(0.451747\pi\)
−0.971074 + 0.238777i \(0.923253\pi\)
\(102\) 0 0
\(103\) 2.14828 + 5.18641i 0.211676 + 0.511032i 0.993681 0.112241i \(-0.0358027\pi\)
−0.782005 + 0.623273i \(0.785803\pi\)
\(104\) 0 0
\(105\) −7.21913 + 17.4285i −0.704515 + 1.70085i
\(106\) 0 0
\(107\) −2.52870 12.7126i −0.244458 1.22897i −0.886654 0.462434i \(-0.846976\pi\)
0.642195 0.766541i \(-0.278024\pi\)
\(108\) 0 0
\(109\) 5.38971 + 8.06626i 0.516240 + 0.772608i 0.994402 0.105664i \(-0.0336967\pi\)
−0.478162 + 0.878272i \(0.658697\pi\)
\(110\) 0 0
\(111\) 5.08545 + 5.08545i 0.482689 + 0.482689i
\(112\) 0 0
\(113\) 4.75634 4.75634i 0.447439 0.447439i −0.447063 0.894502i \(-0.647530\pi\)
0.894502 + 0.447063i \(0.147530\pi\)
\(114\) 0 0
\(115\) −6.74169 + 4.50465i −0.628666 + 0.420061i
\(116\) 0 0
\(117\) 11.9154 2.37013i 1.10158 0.219118i
\(118\) 0 0
\(119\) −6.42293 2.66046i −0.588789 0.243884i
\(120\) 0 0
\(121\) −6.22889 + 2.58009i −0.566262 + 0.234554i
\(122\) 0 0
\(123\) 28.7478 + 19.2087i 2.59210 + 1.73199i
\(124\) 0 0
\(125\) 0.146412 0.736062i 0.0130955 0.0658354i
\(126\) 0 0
\(127\) −0.302397 −0.0268334 −0.0134167 0.999910i \(-0.504271\pi\)
−0.0134167 + 0.999910i \(0.504271\pi\)
\(128\) 0 0
\(129\) 6.58082 0.579409
\(130\) 0 0
\(131\) 1.05063 5.28186i 0.0917938 0.461478i −0.907360 0.420354i \(-0.861906\pi\)
0.999154 0.0411246i \(-0.0130941\pi\)
\(132\) 0 0
\(133\) −7.56673 5.05593i −0.656119 0.438404i
\(134\) 0 0
\(135\) 16.8772 6.99077i 1.45256 0.601669i
\(136\) 0 0
\(137\) 15.9084 + 6.58948i 1.35915 + 0.562978i 0.938824 0.344397i \(-0.111917\pi\)
0.420324 + 0.907374i \(0.361917\pi\)
\(138\) 0 0
\(139\) −11.7049 + 2.32824i −0.992793 + 0.197479i −0.664654 0.747151i \(-0.731421\pi\)
−0.328138 + 0.944630i \(0.606421\pi\)
\(140\) 0 0
\(141\) −7.58783 + 5.07002i −0.639010 + 0.426973i
\(142\) 0 0
\(143\) −3.50343 + 3.50343i −0.292971 + 0.292971i
\(144\) 0 0
\(145\) 5.27222 + 5.27222i 0.437834 + 0.437834i
\(146\) 0 0
\(147\) −3.90511 5.84440i −0.322088 0.482038i
\(148\) 0 0
\(149\) −2.62919 13.2179i −0.215392 1.08285i −0.925498 0.378753i \(-0.876353\pi\)
0.710106 0.704095i \(-0.248647\pi\)
\(150\) 0 0
\(151\) 0.726711 1.75444i 0.0591389 0.142774i −0.891548 0.452926i \(-0.850380\pi\)
0.950687 + 0.310152i \(0.100380\pi\)
\(152\) 0 0
\(153\) 6.32873 + 15.2789i 0.511648 + 1.23523i
\(154\) 0 0
\(155\) 3.14313 4.70402i 0.252462 0.377836i
\(156\) 0 0
\(157\) −18.2962 3.63933i −1.46019 0.290450i −0.599828 0.800129i \(-0.704764\pi\)
−0.860365 + 0.509679i \(0.829764\pi\)
\(158\) 0 0
\(159\) 23.1268i 1.83408i
\(160\) 0 0
\(161\) 5.52037i 0.435066i
\(162\) 0 0
\(163\) 3.25039 + 0.646543i 0.254590 + 0.0506412i 0.320735 0.947169i \(-0.396070\pi\)
−0.0661449 + 0.997810i \(0.521070\pi\)
\(164\) 0 0
\(165\) −10.1676 + 15.2168i −0.791543 + 1.18463i
\(166\) 0 0
\(167\) −0.0933446 0.225354i −0.00722322 0.0174384i 0.920227 0.391386i \(-0.128004\pi\)
−0.927450 + 0.373947i \(0.878004\pi\)
\(168\) 0 0
\(169\) −2.76861 + 6.68401i −0.212970 + 0.514154i
\(170\) 0 0
\(171\) 4.22335 + 21.2322i 0.322968 + 1.62367i
\(172\) 0 0
\(173\) −8.24361 12.3374i −0.626750 0.937998i −0.999948 0.0102459i \(-0.996739\pi\)
0.373197 0.927752i \(-0.378261\pi\)
\(174\) 0 0
\(175\) 7.15875 + 7.15875i 0.541151 + 0.541151i
\(176\) 0 0
\(177\) 17.0942 17.0942i 1.28488 1.28488i
\(178\) 0 0
\(179\) −9.58585 + 6.40506i −0.716480 + 0.478736i −0.859599 0.510969i \(-0.829287\pi\)
0.143119 + 0.989705i \(0.454287\pi\)
\(180\) 0 0
\(181\) −8.33755 + 1.65844i −0.619726 + 0.123271i −0.494961 0.868915i \(-0.664818\pi\)
−0.124765 + 0.992186i \(0.539818\pi\)
\(182\) 0 0
\(183\) −26.3506 10.9148i −1.94789 0.806842i
\(184\) 0 0
\(185\) 7.31172 3.02862i 0.537569 0.222668i
\(186\) 0 0
\(187\) −5.60785 3.74704i −0.410086 0.274011i
\(188\) 0 0
\(189\) 2.42642 12.1985i 0.176496 0.887307i
\(190\) 0 0
\(191\) −6.31207 −0.456725 −0.228363 0.973576i \(-0.573337\pi\)
−0.228363 + 0.973576i \(0.573337\pi\)
\(192\) 0 0
\(193\) −10.5158 −0.756940 −0.378470 0.925614i \(-0.623550\pi\)
−0.378470 + 0.925614i \(0.623550\pi\)
\(194\) 0 0
\(195\) 4.15456 20.8864i 0.297515 1.49571i
\(196\) 0 0
\(197\) 13.0604 + 8.72666i 0.930513 + 0.621749i 0.925708 0.378240i \(-0.123470\pi\)
0.00480529 + 0.999988i \(0.498470\pi\)
\(198\) 0 0
\(199\) −25.4075 + 10.5241i −1.80109 + 0.746037i −0.815085 + 0.579342i \(0.803310\pi\)
−0.986008 + 0.166695i \(0.946690\pi\)
\(200\) 0 0
\(201\) −38.8465 16.0907i −2.74002 1.13495i
\(202\) 0 0
\(203\) 4.97885 0.990354i 0.349447 0.0695092i
\(204\) 0 0
\(205\) 31.6348 21.1377i 2.20947 1.47632i
\(206\) 0 0
\(207\) 9.28566 9.28566i 0.645398 0.645398i
\(208\) 0 0
\(209\) −6.24279 6.24279i −0.431823 0.431823i
\(210\) 0 0
\(211\) −12.5416 18.7698i −0.863399 1.29217i −0.955071 0.296378i \(-0.904221\pi\)
0.0916719 0.995789i \(-0.470779\pi\)
\(212\) 0 0
\(213\) 3.84542 + 19.3322i 0.263484 + 1.32462i
\(214\) 0 0
\(215\) 2.77128 6.69045i 0.189000 0.456285i
\(216\) 0 0
\(217\) −1.47404 3.55864i −0.100064 0.241576i
\(218\) 0 0
\(219\) 5.74193 8.59341i 0.388004 0.580689i
\(220\) 0 0
\(221\) 7.69726 + 1.53108i 0.517774 + 0.102992i
\(222\) 0 0
\(223\) 8.11318i 0.543298i −0.962396 0.271649i \(-0.912431\pi\)
0.962396 0.271649i \(-0.0875690\pi\)
\(224\) 0 0
\(225\) 24.0831i 1.60554i
\(226\) 0 0
\(227\) 11.3933 + 2.26626i 0.756197 + 0.150417i 0.558109 0.829767i \(-0.311527\pi\)
0.198087 + 0.980184i \(0.436527\pi\)
\(228\) 0 0
\(229\) 12.2867 18.3883i 0.811926 1.21513i −0.161668 0.986845i \(-0.551687\pi\)
0.973594 0.228288i \(-0.0733127\pi\)
\(230\) 0 0
\(231\) 4.76829 + 11.5117i 0.313731 + 0.757413i
\(232\) 0 0
\(233\) 0.609526 1.47153i 0.0399314 0.0964028i −0.902656 0.430364i \(-0.858385\pi\)
0.942587 + 0.333961i \(0.108385\pi\)
\(234\) 0 0
\(235\) 1.95915 + 9.84930i 0.127801 + 0.642497i
\(236\) 0 0
\(237\) 24.4225 + 36.5508i 1.58641 + 2.37423i
\(238\) 0 0
\(239\) 1.07033 + 1.07033i 0.0692337 + 0.0692337i 0.740876 0.671642i \(-0.234411\pi\)
−0.671642 + 0.740876i \(0.734411\pi\)
\(240\) 0 0
\(241\) 11.5889 11.5889i 0.746509 0.746509i −0.227313 0.973822i \(-0.572994\pi\)
0.973822 + 0.227313i \(0.0729940\pi\)
\(242\) 0 0
\(243\) 11.2304 7.50390i 0.720429 0.481375i
\(244\) 0 0
\(245\) −7.58626 + 1.50900i −0.484669 + 0.0964066i
\(246\) 0 0
\(247\) 9.49121 + 3.93139i 0.603911 + 0.250148i
\(248\) 0 0
\(249\) −17.5997 + 7.29002i −1.11533 + 0.461986i
\(250\) 0 0
\(251\) 3.20465 + 2.14128i 0.202276 + 0.135156i 0.652583 0.757717i \(-0.273685\pi\)
−0.450307 + 0.892874i \(0.648685\pi\)
\(252\) 0 0
\(253\) −1.04481 + 5.25260i −0.0656865 + 0.330228i
\(254\) 0 0
\(255\) 28.9889 1.81535
\(256\) 0 0
\(257\) −12.4093 −0.774071 −0.387035 0.922065i \(-0.626501\pi\)
−0.387035 + 0.922065i \(0.626501\pi\)
\(258\) 0 0
\(259\) 1.05120 5.28475i 0.0653185 0.328378i
\(260\) 0 0
\(261\) −10.0406 6.70893i −0.621499 0.415272i
\(262\) 0 0
\(263\) −1.63628 + 0.677769i −0.100897 + 0.0417930i −0.432561 0.901605i \(-0.642390\pi\)
0.331664 + 0.943398i \(0.392390\pi\)
\(264\) 0 0
\(265\) 23.5121 + 9.73904i 1.44434 + 0.598264i
\(266\) 0 0
\(267\) 22.5989 4.49521i 1.38303 0.275102i
\(268\) 0 0
\(269\) −26.2384 + 17.5320i −1.59978 + 1.06894i −0.648238 + 0.761438i \(0.724494\pi\)
−0.951547 + 0.307505i \(0.900506\pi\)
\(270\) 0 0
\(271\) 1.58765 1.58765i 0.0964427 0.0964427i −0.657239 0.753682i \(-0.728276\pi\)
0.753682 + 0.657239i \(0.228276\pi\)
\(272\) 0 0
\(273\) −10.2523 10.2523i −0.620497 0.620497i
\(274\) 0 0
\(275\) 5.45661 + 8.16640i 0.329046 + 0.492452i
\(276\) 0 0
\(277\) −0.970436 4.87871i −0.0583079 0.293133i 0.940618 0.339466i \(-0.110246\pi\)
−0.998926 + 0.0463330i \(0.985246\pi\)
\(278\) 0 0
\(279\) −3.50646 + 8.46533i −0.209926 + 0.506806i
\(280\) 0 0
\(281\) −0.758051 1.83010i −0.0452216 0.109175i 0.899655 0.436602i \(-0.143818\pi\)
−0.944876 + 0.327428i \(0.893818\pi\)
\(282\) 0 0
\(283\) −5.42628 + 8.12100i −0.322559 + 0.482744i −0.956944 0.290274i \(-0.906254\pi\)
0.634385 + 0.773018i \(0.281254\pi\)
\(284\) 0 0
\(285\) 37.2177 + 7.40306i 2.20459 + 0.438519i
\(286\) 0 0
\(287\) 25.9039i 1.52906i
\(288\) 0 0
\(289\) 6.31674i 0.371573i
\(290\) 0 0
\(291\) −23.6476 4.70380i −1.38625 0.275742i
\(292\) 0 0
\(293\) 6.65651 9.96217i 0.388877 0.581996i −0.584446 0.811432i \(-0.698688\pi\)
0.973324 + 0.229436i \(0.0736883\pi\)
\(294\) 0 0
\(295\) −10.1804 24.5776i −0.592725 1.43096i
\(296\) 0 0
\(297\) 4.61746 11.1475i 0.267932 0.646845i
\(298\) 0 0
\(299\) −1.21576 6.11205i −0.0703093 0.353469i
\(300\) 0 0
\(301\) −2.73921 4.09952i −0.157885 0.236292i
\(302\) 0 0
\(303\) −29.7792 29.7792i −1.71077 1.71077i
\(304\) 0 0
\(305\) −22.1932 + 22.1932i −1.27078 + 1.27078i
\(306\) 0 0
\(307\) 9.59801 6.41318i 0.547787 0.366020i −0.250669 0.968073i \(-0.580650\pi\)
0.798456 + 0.602053i \(0.205650\pi\)
\(308\) 0 0
\(309\) −15.6309 + 3.10919i −0.889213 + 0.176875i
\(310\) 0 0
\(311\) 21.5704 + 8.93477i 1.22315 + 0.506644i 0.898409 0.439160i \(-0.144724\pi\)
0.324738 + 0.945804i \(0.394724\pi\)
\(312\) 0 0
\(313\) −1.66011 + 0.687642i −0.0938352 + 0.0388678i −0.429107 0.903254i \(-0.641172\pi\)
0.335272 + 0.942121i \(0.391172\pi\)
\(314\) 0 0
\(315\) −27.9549 18.6788i −1.57508 1.05243i
\(316\) 0 0
\(317\) −2.90261 + 14.5924i −0.163027 + 0.819590i 0.809558 + 0.587040i \(0.199707\pi\)
−0.972584 + 0.232550i \(0.925293\pi\)
\(318\) 0 0
\(319\) 4.92478 0.275735
\(320\) 0 0
\(321\) 36.7977 2.05385
\(322\) 0 0
\(323\) −2.72825 + 13.7158i −0.151804 + 0.763169i
\(324\) 0 0
\(325\) −9.50261 6.34944i −0.527110 0.352204i
\(326\) 0 0
\(327\) −25.4449 + 10.5396i −1.40711 + 0.582843i
\(328\) 0 0
\(329\) 6.31673 + 2.61648i 0.348253 + 0.144251i
\(330\) 0 0
\(331\) −5.51285 + 1.09657i −0.303014 + 0.0602732i −0.344255 0.938876i \(-0.611869\pi\)
0.0412418 + 0.999149i \(0.486869\pi\)
\(332\) 0 0
\(333\) −10.6575 + 7.12113i −0.584029 + 0.390236i
\(334\) 0 0
\(335\) −32.7176 + 32.7176i −1.78755 + 1.78755i
\(336\) 0 0
\(337\) 2.17791 + 2.17791i 0.118638 + 0.118638i 0.763933 0.645295i \(-0.223266\pi\)
−0.645295 + 0.763933i \(0.723266\pi\)
\(338\) 0 0
\(339\) 10.6093 + 15.8779i 0.576218 + 0.862371i
\(340\) 0 0
\(341\) −0.729016 3.66501i −0.0394784 0.198471i
\(342\) 0 0
\(343\) −7.71305 + 18.6210i −0.416466 + 1.00544i
\(344\) 0 0
\(345\) −8.80890 21.2666i −0.474255 1.14495i
\(346\) 0 0
\(347\) −5.32825 + 7.97430i −0.286036 + 0.428083i −0.946465 0.322805i \(-0.895374\pi\)
0.660430 + 0.750888i \(0.270374\pi\)
\(348\) 0 0
\(349\) −1.63592 0.325405i −0.0875689 0.0174185i 0.151111 0.988517i \(-0.451715\pi\)
−0.238680 + 0.971098i \(0.576715\pi\)
\(350\) 0 0
\(351\) 14.0403i 0.749414i
\(352\) 0 0
\(353\) 5.92592i 0.315405i 0.987487 + 0.157702i \(0.0504087\pi\)
−0.987487 + 0.157702i \(0.949591\pi\)
\(354\) 0 0
\(355\) 21.2737 + 4.23159i 1.12909 + 0.224590i
\(356\) 0 0
\(357\) 10.9652 16.4106i 0.580340 0.868539i
\(358\) 0 0
\(359\) 12.0537 + 29.1002i 0.636170 + 1.53585i 0.831743 + 0.555161i \(0.187343\pi\)
−0.195573 + 0.980689i \(0.562657\pi\)
\(360\) 0 0
\(361\) 0.265606 0.641230i 0.0139793 0.0337489i
\(362\) 0 0
\(363\) −3.73414 18.7728i −0.195991 0.985315i
\(364\) 0 0
\(365\) −6.31856 9.45639i −0.330729 0.494970i
\(366\) 0 0
\(367\) 18.9541 + 18.9541i 0.989396 + 0.989396i 0.999944 0.0105483i \(-0.00335770\pi\)
−0.0105483 + 0.999944i \(0.503358\pi\)
\(368\) 0 0
\(369\) −43.5722 + 43.5722i −2.26828 + 2.26828i
\(370\) 0 0
\(371\) 14.4068 9.62634i 0.747966 0.499775i
\(372\) 0 0
\(373\) −21.0597 + 4.18903i −1.09043 + 0.216899i −0.707383 0.706830i \(-0.750124\pi\)
−0.383044 + 0.923730i \(0.625124\pi\)
\(374\) 0 0
\(375\) 1.96841 + 0.815342i 0.101648 + 0.0421041i
\(376\) 0 0
\(377\) −5.29437 + 2.19300i −0.272674 + 0.112945i
\(378\) 0 0
\(379\) −3.08942 2.06428i −0.158693 0.106035i 0.473688 0.880693i \(-0.342922\pi\)
−0.632381 + 0.774657i \(0.717922\pi\)
\(380\) 0 0
\(381\) 0.167484 0.841999i 0.00858046 0.0431369i
\(382\) 0 0
\(383\) −16.3597 −0.835942 −0.417971 0.908460i \(-0.637259\pi\)
−0.417971 + 0.908460i \(0.637259\pi\)
\(384\) 0 0
\(385\) 13.7115 0.698800
\(386\) 0 0
\(387\) −2.28813 + 11.5032i −0.116312 + 0.584742i
\(388\) 0 0
\(389\) 21.6591 + 14.4722i 1.09816 + 0.733768i 0.966279 0.257498i \(-0.0828980\pi\)
0.131882 + 0.991265i \(0.457898\pi\)
\(390\) 0 0
\(391\) 7.83736 3.24634i 0.396352 0.164174i
\(392\) 0 0
\(393\) 14.1250 + 5.85076i 0.712511 + 0.295132i
\(394\) 0 0
\(395\) 47.4444 9.43728i 2.38719 0.474841i
\(396\) 0 0
\(397\) 3.68511 2.46232i 0.184951 0.123580i −0.459648 0.888101i \(-0.652024\pi\)
0.644599 + 0.764521i \(0.277024\pi\)
\(398\) 0 0
\(399\) 18.2686 18.2686i 0.914576 0.914576i
\(400\) 0 0
\(401\) 8.30297 + 8.30297i 0.414630 + 0.414630i 0.883348 0.468718i \(-0.155284\pi\)
−0.468718 + 0.883348i \(0.655284\pi\)
\(402\) 0 0
\(403\) 2.41575 + 3.61543i 0.120337 + 0.180097i
\(404\) 0 0
\(405\) 0.866379 + 4.35558i 0.0430507 + 0.216431i
\(406\) 0 0
\(407\) 2.00042 4.82945i 0.0991573 0.239387i
\(408\) 0 0
\(409\) 12.9503 + 31.2648i 0.640350 + 1.54594i 0.826207 + 0.563366i \(0.190494\pi\)
−0.185857 + 0.982577i \(0.559506\pi\)
\(410\) 0 0
\(411\) −27.1588 + 40.6460i −1.33964 + 2.00492i
\(412\) 0 0
\(413\) −17.7641 3.53351i −0.874117 0.173873i
\(414\) 0 0
\(415\) 20.9628i 1.02902i
\(416\) 0 0
\(417\) 33.8806i 1.65914i
\(418\) 0 0
\(419\) −15.5325 3.08960i −0.758811 0.150937i −0.199503 0.979897i \(-0.563933\pi\)
−0.559307 + 0.828960i \(0.688933\pi\)
\(420\) 0 0
\(421\) 1.42193 2.12807i 0.0693006 0.103716i −0.795201 0.606346i \(-0.792634\pi\)
0.864501 + 0.502631i \(0.167634\pi\)
\(422\) 0 0
\(423\) −6.22410 15.0263i −0.302626 0.730604i
\(424\) 0 0
\(425\) 5.95357 14.3732i 0.288791 0.697203i
\(426\) 0 0
\(427\) 4.16885 + 20.9582i 0.201745 + 1.01424i
\(428\) 0 0
\(429\) −7.81460 11.6954i −0.377292 0.564658i
\(430\) 0 0
\(431\) −13.2163 13.2163i −0.636606 0.636606i 0.313111 0.949717i \(-0.398629\pi\)
−0.949717 + 0.313111i \(0.898629\pi\)
\(432\) 0 0
\(433\) −5.42276 + 5.42276i −0.260601 + 0.260601i −0.825298 0.564697i \(-0.808993\pi\)
0.564697 + 0.825298i \(0.308993\pi\)
\(434\) 0 0
\(435\) −17.6001 + 11.7600i −0.843860 + 0.563849i
\(436\) 0 0
\(437\) 10.8911 2.16638i 0.520993 0.103632i
\(438\) 0 0
\(439\) 15.4472 + 6.39844i 0.737255 + 0.305381i 0.719530 0.694462i \(-0.244357\pi\)
0.0177256 + 0.999843i \(0.494357\pi\)
\(440\) 0 0
\(441\) 11.5738 4.79401i 0.551132 0.228286i
\(442\) 0 0
\(443\) −8.77140 5.86086i −0.416742 0.278458i 0.329471 0.944166i \(-0.393130\pi\)
−0.746213 + 0.665708i \(0.768130\pi\)
\(444\) 0 0
\(445\) 4.94664 24.8684i 0.234493 1.17888i
\(446\) 0 0
\(447\) 38.2601 1.80964
\(448\) 0 0
\(449\) 11.0927 0.523498 0.261749 0.965136i \(-0.415701\pi\)
0.261749 + 0.965136i \(0.415701\pi\)
\(450\) 0 0
\(451\) 4.90267 24.6474i 0.230857 1.16060i
\(452\) 0 0
\(453\) 4.48258 + 2.99516i 0.210610 + 0.140725i
\(454\) 0 0
\(455\) −14.7405 + 6.10570i −0.691044 + 0.286240i
\(456\) 0 0
\(457\) 20.2978 + 8.40761i 0.949490 + 0.393292i 0.803039 0.595926i \(-0.203215\pi\)
0.146451 + 0.989218i \(0.453215\pi\)
\(458\) 0 0
\(459\) −18.7452 + 3.72866i −0.874953 + 0.174039i
\(460\) 0 0
\(461\) 19.1714 12.8099i 0.892903 0.596619i −0.0222379 0.999753i \(-0.507079\pi\)
0.915141 + 0.403134i \(0.132079\pi\)
\(462\) 0 0
\(463\) 9.19024 9.19024i 0.427107 0.427107i −0.460535 0.887642i \(-0.652342\pi\)
0.887642 + 0.460535i \(0.152342\pi\)
\(464\) 0 0
\(465\) 11.3571 + 11.3571i 0.526673 + 0.526673i
\(466\) 0 0
\(467\) −0.319653 0.478395i −0.0147918 0.0221375i 0.824000 0.566590i \(-0.191738\pi\)
−0.838792 + 0.544452i \(0.816738\pi\)
\(468\) 0 0
\(469\) 6.14580 + 30.8970i 0.283787 + 1.42669i
\(470\) 0 0
\(471\) 20.2668 48.9284i 0.933845 2.25450i
\(472\) 0 0
\(473\) −1.83045 4.41910i −0.0841642 0.203190i
\(474\) 0 0
\(475\) 11.3141 16.9328i 0.519128 0.776930i
\(476\) 0 0
\(477\) −40.4255 8.04114i −1.85096 0.368178i
\(478\) 0 0
\(479\) 33.0010i 1.50786i 0.656957 + 0.753928i \(0.271843\pi\)
−0.656957 + 0.753928i \(0.728157\pi\)
\(480\) 0 0
\(481\) 6.08268i 0.277346i
\(482\) 0 0
\(483\) −15.3710 3.05748i −0.699404 0.139120i
\(484\) 0 0
\(485\) −14.7405 + 22.0607i −0.669332 + 1.00173i
\(486\) 0 0
\(487\) −14.4792 34.9559i −0.656116 1.58400i −0.803753 0.594963i \(-0.797167\pi\)
0.147637 0.989042i \(-0.452833\pi\)
\(488\) 0 0
\(489\) −3.60049 + 8.69235i −0.162820 + 0.393081i
\(490\) 0 0
\(491\) −4.13764 20.8013i −0.186729 0.938750i −0.954542 0.298076i \(-0.903655\pi\)
0.767813 0.640674i \(-0.221345\pi\)
\(492\) 0 0
\(493\) −4.33391 6.48615i −0.195189 0.292122i
\(494\) 0 0
\(495\) −23.0637 23.0637i −1.03663 1.03663i
\(496\) 0 0
\(497\) 10.4424 10.4424i 0.468404 0.468404i
\(498\) 0 0
\(499\) −34.2079 + 22.8570i −1.53135 + 1.02322i −0.549079 + 0.835770i \(0.685021\pi\)
−0.982275 + 0.187448i \(0.939979\pi\)
\(500\) 0 0
\(501\) 0.679177 0.135097i 0.0303434 0.00603568i
\(502\) 0 0
\(503\) 16.6755 + 6.90723i 0.743525 + 0.307978i 0.722097 0.691792i \(-0.243179\pi\)
0.0214284 + 0.999770i \(0.493179\pi\)
\(504\) 0 0
\(505\) −42.8157 + 17.7349i −1.90528 + 0.789191i
\(506\) 0 0
\(507\) −17.0776 11.4109i −0.758444 0.506776i
\(508\) 0 0
\(509\) 4.55107 22.8798i 0.201722 1.01413i −0.738678 0.674058i \(-0.764550\pi\)
0.940401 0.340069i \(-0.110450\pi\)
\(510\) 0 0
\(511\) −7.74328 −0.342543
\(512\) 0 0
\(513\) −25.0185 −1.10459
\(514\) 0 0
\(515\) −3.42142 + 17.2007i −0.150766 + 0.757952i
\(516\) 0 0
\(517\) 5.51513 + 3.68509i 0.242555 + 0.162070i
\(518\) 0 0
\(519\) 38.9183 16.1205i 1.70832 0.707610i
\(520\) 0 0
\(521\) −25.8128 10.6920i −1.13088 0.468426i −0.262801 0.964850i \(-0.584646\pi\)
−0.868080 + 0.496424i \(0.834646\pi\)
\(522\) 0 0
\(523\) 3.91430 0.778602i 0.171160 0.0340459i −0.108766 0.994067i \(-0.534690\pi\)
0.279926 + 0.960021i \(0.409690\pi\)
\(524\) 0 0
\(525\) −23.8978 + 15.9680i −1.04299 + 0.696901i
\(526\) 0 0
\(527\) −4.18543 + 4.18543i −0.182320 + 0.182320i
\(528\) 0 0
\(529\) 11.5004 + 11.5004i 0.500015 + 0.500015i
\(530\) 0 0
\(531\) 23.9370 + 35.8242i 1.03878 + 1.55464i
\(532\) 0 0
\(533\) 5.70486 + 28.6802i 0.247105 + 1.24228i
\(534\) 0 0
\(535\) 15.4960 37.4107i 0.669952 1.61741i
\(536\) 0 0
\(537\) −12.5252 30.2384i −0.540501 1.30488i
\(538\) 0 0
\(539\) −2.83838 + 4.24794i −0.122258 + 0.182972i
\(540\) 0 0
\(541\) −23.5841 4.69116i −1.01396 0.201689i −0.339974 0.940435i \(-0.610418\pi\)
−0.673984 + 0.738746i \(0.735418\pi\)
\(542\) 0 0
\(543\) 24.1337i 1.03568i
\(544\) 0 0
\(545\) 30.3072i 1.29822i
\(546\) 0 0
\(547\) 18.9505 + 3.76948i 0.810263 + 0.161171i 0.582800 0.812615i \(-0.301957\pi\)
0.227463 + 0.973787i \(0.426957\pi\)
\(548\) 0 0
\(549\) 28.2409 42.2656i 1.20529 1.80385i
\(550\) 0 0
\(551\) −3.90773 9.43410i −0.166475 0.401906i
\(552\) 0 0
\(553\) 12.6037 30.4279i 0.535962 1.29393i
\(554\) 0 0
\(555\) 4.38329 + 22.0363i 0.186060 + 0.935387i
\(556\) 0 0
\(557\) 9.35372 + 13.9988i 0.396330 + 0.593149i 0.974944 0.222452i \(-0.0714062\pi\)
−0.578614 + 0.815602i \(0.696406\pi\)
\(558\) 0 0
\(559\) 3.93564 + 3.93564i 0.166460 + 0.166460i
\(560\) 0 0
\(561\) 13.5392 13.5392i 0.571627 0.571627i
\(562\) 0 0
\(563\) −33.2172 + 22.1950i −1.39994 + 0.935409i −0.400119 + 0.916463i \(0.631031\pi\)
−0.999821 + 0.0189460i \(0.993969\pi\)
\(564\) 0 0
\(565\) 20.6102 4.09962i 0.867077 0.172472i
\(566\) 0 0
\(567\) 2.79340 + 1.15706i 0.117312 + 0.0485922i
\(568\) 0 0
\(569\) 14.9275 6.18318i 0.625794 0.259212i −0.0471707 0.998887i \(-0.515020\pi\)
0.672965 + 0.739674i \(0.265020\pi\)
\(570\) 0 0
\(571\) 10.6636 + 7.12517i 0.446257 + 0.298179i 0.758320 0.651882i \(-0.226020\pi\)
−0.312064 + 0.950061i \(0.601020\pi\)
\(572\) 0 0
\(573\) 3.49596 17.5754i 0.146046 0.734223i
\(574\) 0 0
\(575\) −12.3535 −0.515175
\(576\) 0 0
\(577\) −18.6883 −0.778002 −0.389001 0.921237i \(-0.627180\pi\)
−0.389001 + 0.921237i \(0.627180\pi\)
\(578\) 0 0
\(579\) 5.82419 29.2802i 0.242045 1.21684i
\(580\) 0 0
\(581\) 11.8670 + 7.92928i 0.492327 + 0.328962i
\(582\) 0 0
\(583\) 15.5299 6.43271i 0.643184 0.266416i
\(584\) 0 0
\(585\) 35.0648 + 14.5243i 1.44975 + 0.600506i
\(586\) 0 0
\(587\) −16.9938 + 3.38028i −0.701409 + 0.139519i −0.532896 0.846180i \(-0.678896\pi\)
−0.168513 + 0.985699i \(0.553896\pi\)
\(588\) 0 0
\(589\) −6.44237 + 4.30465i −0.265453 + 0.177370i
\(590\) 0 0
\(591\) −31.5322 + 31.5322i −1.29706 + 1.29706i
\(592\) 0 0
\(593\) −5.79685 5.79685i −0.238048 0.238048i 0.577993 0.816042i \(-0.303836\pi\)
−0.816042 + 0.577993i \(0.803836\pi\)
\(594\) 0 0
\(595\) −12.0664 18.0586i −0.494673 0.740330i
\(596\) 0 0
\(597\) −15.2315 76.5739i −0.623384 3.13396i
\(598\) 0 0
\(599\) 6.07554 14.6677i 0.248240 0.599304i −0.749815 0.661648i \(-0.769857\pi\)
0.998055 + 0.0623435i \(0.0198574\pi\)
\(600\) 0 0
\(601\) −9.70092 23.4201i −0.395709 0.955325i −0.988672 0.150095i \(-0.952042\pi\)
0.592963 0.805230i \(-0.297958\pi\)
\(602\) 0 0
\(603\) 41.6333 62.3087i 1.69544 2.53741i
\(604\) 0 0
\(605\) −20.6580 4.10914i −0.839868 0.167060i
\(606\) 0 0
\(607\) 0.795354i 0.0322824i 0.999870 + 0.0161412i \(0.00513813\pi\)
−0.999870 + 0.0161412i \(0.994862\pi\)
\(608\) 0 0
\(609\) 14.4117i 0.583991i
\(610\) 0 0
\(611\) −7.57000 1.50577i −0.306249 0.0609168i
\(612\) 0 0
\(613\) −2.28756 + 3.42358i −0.0923939 + 0.138277i −0.874787 0.484508i \(-0.838999\pi\)
0.782393 + 0.622785i \(0.213999\pi\)
\(614\) 0 0
\(615\) 41.3350 + 99.7915i 1.66679 + 4.02398i
\(616\) 0 0
\(617\) −16.1612 + 39.0166i −0.650626 + 1.57075i 0.161246 + 0.986914i \(0.448449\pi\)
−0.811872 + 0.583836i \(0.801551\pi\)
\(618\) 0 0
\(619\) −2.87121 14.4345i −0.115404 0.580173i −0.994606 0.103725i \(-0.966924\pi\)
0.879202 0.476448i \(-0.158076\pi\)
\(620\) 0 0
\(621\) 8.43154 + 12.6187i 0.338346 + 0.506370i
\(622\) 0 0
\(623\) −12.2069 12.2069i −0.489059 0.489059i
\(624\) 0 0
\(625\) 18.4862 18.4862i 0.739448 0.739448i
\(626\) 0 0
\(627\) 20.8401 13.9249i 0.832274 0.556107i
\(628\) 0 0
\(629\) −8.12102 + 1.61537i −0.323806 + 0.0644090i
\(630\) 0 0
\(631\) −35.2699 14.6093i −1.40407 0.581586i −0.453268 0.891374i \(-0.649742\pi\)
−0.950806 + 0.309788i \(0.899742\pi\)
\(632\) 0 0
\(633\) 59.2091 24.5252i 2.35335 0.974790i
\(634\) 0 0
\(635\) −0.785496 0.524852i −0.0311715 0.0208281i
\(636\) 0 0
\(637\) 1.15979 5.83067i 0.0459526 0.231020i
\(638\) 0 0
\(639\) −35.1296 −1.38971
\(640\) 0 0
\(641\) 11.8042 0.466237 0.233119 0.972448i \(-0.425107\pi\)
0.233119 + 0.972448i \(0.425107\pi\)
\(642\) 0 0
\(643\) 2.84656 14.3106i 0.112257 0.564355i −0.883188 0.469019i \(-0.844607\pi\)
0.995445 0.0953360i \(-0.0303926\pi\)
\(644\) 0 0
\(645\) 17.0941 + 11.4219i 0.673080 + 0.449737i
\(646\) 0 0
\(647\) 6.76078 2.80041i 0.265794 0.110095i −0.245807 0.969319i \(-0.579053\pi\)
0.511600 + 0.859224i \(0.329053\pi\)
\(648\) 0 0
\(649\) −16.2337 6.72422i −0.637229 0.263949i
\(650\) 0 0
\(651\) 10.7251 2.13336i 0.420351 0.0836130i
\(652\) 0 0
\(653\) 9.41350 6.28990i 0.368379 0.246143i −0.357578 0.933883i \(-0.616397\pi\)
0.725957 + 0.687741i \(0.241397\pi\)
\(654\) 0 0
\(655\) 11.8965 11.8965i 0.464833 0.464833i
\(656\) 0 0
\(657\) 13.0248 + 13.0248i 0.508144 + 0.508144i
\(658\) 0 0
\(659\) 6.27371 + 9.38927i 0.244389 + 0.365754i 0.933303 0.359089i \(-0.116912\pi\)
−0.688914 + 0.724843i \(0.741912\pi\)
\(660\) 0 0
\(661\) −0.0233980 0.117629i −0.000910075 0.00457526i 0.980328 0.197377i \(-0.0632424\pi\)
−0.981238 + 0.192802i \(0.938242\pi\)
\(662\) 0 0
\(663\) −8.52632 + 20.5843i −0.331135 + 0.799430i
\(664\) 0 0
\(665\) −10.8798 26.2662i −0.421901 1.01856i
\(666\) 0 0
\(667\) −3.44136 + 5.15036i −0.133250 + 0.199423i
\(668\) 0 0
\(669\) 22.5904 + 4.49352i 0.873396 + 0.173729i
\(670\) 0 0
\(671\) 20.7306i 0.800298i
\(672\) 0 0
\(673\) 41.1783i 1.58731i −0.608371 0.793653i \(-0.708177\pi\)
0.608371 0.793653i \(-0.291823\pi\)
\(674\) 0 0
\(675\) 27.2977 + 5.42984i 1.05069 + 0.208995i
\(676\) 0 0
\(677\) −13.0241 + 19.4920i −0.500557 + 0.749137i −0.992599 0.121435i \(-0.961251\pi\)
0.492042 + 0.870571i \(0.336251\pi\)
\(678\) 0 0
\(679\) 6.91289 + 16.6892i 0.265292 + 0.640472i
\(680\) 0 0
\(681\) −12.6204 + 30.4683i −0.483615 + 1.16755i
\(682\) 0 0
\(683\) −2.94439 14.8024i −0.112664 0.566399i −0.995341 0.0964191i \(-0.969261\pi\)
0.882677 0.469980i \(-0.155739\pi\)
\(684\) 0 0
\(685\) 29.8862 + 44.7278i 1.14189 + 1.70896i
\(686\) 0 0
\(687\) 44.3956 + 44.3956i 1.69380 + 1.69380i
\(688\) 0 0
\(689\) −13.8309 + 13.8309i −0.526917 + 0.526917i
\(690\) 0 0
\(691\) 23.4953 15.6991i 0.893805 0.597221i −0.0215953 0.999767i \(-0.506875\pi\)
0.915400 + 0.402546i \(0.131875\pi\)
\(692\) 0 0
\(693\) −21.7803 + 4.33236i −0.827364 + 0.164573i
\(694\) 0 0
\(695\) −34.4451 14.2676i −1.30658 0.541202i
\(696\) 0 0
\(697\) −36.7761 + 15.2332i −1.39299 + 0.576997i
\(698\) 0 0
\(699\) 3.75975 + 2.51218i 0.142207 + 0.0950194i
\(700\) 0 0
\(701\) 1.68037 8.44781i 0.0634668 0.319069i −0.935990 0.352027i \(-0.885493\pi\)
0.999457 + 0.0329575i \(0.0104926\pi\)
\(702\) 0 0
\(703\) −10.8388 −0.408792
\(704\) 0 0
\(705\) −28.5096 −1.07373
\(706\) 0 0
\(707\) −6.15559 + 30.9462i −0.231505 + 1.16385i
\(708\) 0 0
\(709\) −9.70687 6.48592i −0.364549 0.243584i 0.359782 0.933036i \(-0.382851\pi\)
−0.724331 + 0.689453i \(0.757851\pi\)
\(710\) 0 0
\(711\) −72.3823 + 29.9817i −2.71455 + 1.12440i
\(712\) 0 0
\(713\) 4.34231 + 1.79865i 0.162621 + 0.0673598i
\(714\) 0 0
\(715\) −15.1811 + 3.01970i −0.567739 + 0.112930i
\(716\) 0 0
\(717\) −3.57303 + 2.38743i −0.133437 + 0.0891600i
\(718\) 0 0
\(719\) 22.6553 22.6553i 0.844899 0.844899i −0.144592 0.989491i \(-0.546187\pi\)
0.989491 + 0.144592i \(0.0461871\pi\)
\(720\) 0 0
\(721\) 8.44310 + 8.44310i 0.314438 + 0.314438i
\(722\) 0 0
\(723\) 25.8498 + 38.6869i 0.961364 + 1.43878i
\(724\) 0 0
\(725\) 2.21621 + 11.1416i 0.0823080 + 0.413790i
\(726\) 0 0
\(727\) −14.7776 + 35.6764i −0.548072 + 1.32316i 0.370839 + 0.928697i \(0.379070\pi\)
−0.918911 + 0.394466i \(0.870930\pi\)
\(728\) 0 0
\(729\) 16.3059 + 39.3660i 0.603923 + 1.45800i
\(730\) 0 0
\(731\) −4.20931 + 6.29968i −0.155687 + 0.233002i
\(732\) 0 0
\(733\) 12.4550 + 2.47746i 0.460037 + 0.0915071i 0.419668 0.907678i \(-0.362147\pi\)
0.0403693 + 0.999185i \(0.487147\pi\)
\(734\) 0 0
\(735\) 21.9591i 0.809972i
\(736\) 0 0
\(737\) 30.5615i 1.12575i
\(738\) 0 0
\(739\) −0.370926 0.0737817i −0.0136447 0.00271410i 0.188263 0.982119i \(-0.439714\pi\)
−0.201908 + 0.979405i \(0.564714\pi\)
\(740\) 0 0
\(741\) −16.2034 + 24.2500i −0.595245 + 0.890847i
\(742\) 0 0
\(743\) 10.5484 + 25.4660i 0.386983 + 0.934259i 0.990576 + 0.136966i \(0.0437351\pi\)
−0.603593 + 0.797292i \(0.706265\pi\)
\(744\) 0 0
\(745\) 16.1119 38.8975i 0.590294 1.42510i
\(746\) 0 0
\(747\) −6.62354 33.2988i −0.242343 1.21834i
\(748\) 0 0
\(749\) −15.3167 22.9231i −0.559661 0.837591i
\(750\) 0 0
\(751\) −33.7123 33.7123i −1.23018 1.23018i −0.963894 0.266286i \(-0.914203\pi\)
−0.266286 0.963894i \(-0.585797\pi\)
\(752\) 0 0
\(753\) −7.73712 + 7.73712i −0.281956 + 0.281956i
\(754\) 0 0
\(755\) 4.93274 3.29595i 0.179521 0.119952i
\(756\) 0 0
\(757\) 0.880215 0.175086i 0.0319920 0.00636360i −0.179068 0.983837i \(-0.557308\pi\)
0.211060 + 0.977473i \(0.432308\pi\)
\(758\) 0 0
\(759\) −14.0467 5.81835i −0.509864 0.211193i
\(760\) 0 0
\(761\) 25.8419 10.7040i 0.936766 0.388021i 0.138525 0.990359i \(-0.455764\pi\)
0.798241 + 0.602338i \(0.205764\pi\)
\(762\) 0 0
\(763\) 17.1569 + 11.4639i 0.621121 + 0.415020i
\(764\) 0 0
\(765\) −10.0794 + 50.6724i −0.364420 + 1.83206i
\(766\) 0 0
\(767\) 20.4463 0.738273
\(768\) 0 0
\(769\) −30.4304 −1.09735 −0.548674 0.836036i \(-0.684867\pi\)
−0.548674 + 0.836036i \(0.684867\pi\)
\(770\) 0 0
\(771\) 6.87294 34.5526i 0.247523 1.24438i
\(772\) 0 0
\(773\) −7.79837 5.21070i −0.280488 0.187416i 0.407366 0.913265i \(-0.366447\pi\)
−0.687854 + 0.725849i \(0.741447\pi\)
\(774\) 0 0
\(775\) 7.96352 3.29860i 0.286058 0.118489i
\(776\) 0 0
\(777\) 14.1327 + 5.85396i 0.507008 + 0.210010i
\(778\) 0 0
\(779\) −51.1056 + 10.1655i −1.83105 + 0.364218i
\(780\) 0 0
\(781\) 11.9122 7.95949i 0.426253 0.284813i
\(782\) 0 0
\(783\) 9.86823 9.86823i 0.352662 0.352662i
\(784\) 0 0
\(785\) −41.2089 41.2089i −1.47081 1.47081i
\(786\) 0 0
\(787\) −19.4919 29.1717i −0.694813 1.03986i −0.996262 0.0863785i \(-0.972471\pi\)
0.301450 0.953482i \(-0.402529\pi\)
\(788\) 0 0
\(789\) −0.980928 4.93146i −0.0349220 0.175565i
\(790\) 0 0
\(791\) 5.47512 13.2181i 0.194673 0.469982i
\(792\) 0 0
\(793\) −9.23134 22.2864i −0.327815 0.791414i
\(794\) 0 0
\(795\) −40.1398 + 60.0734i −1.42361 + 2.13058i
\(796\) 0 0
\(797\) −11.1662 2.22109i −0.395526 0.0786751i −0.00667994 0.999978i \(-0.502126\pi\)
−0.388846 + 0.921303i \(0.627126\pi\)
\(798\) 0 0
\(799\) 10.5066i 0.371698i
\(800\) 0 0
\(801\) 41.0658i 1.45099i
\(802\) 0 0
\(803\) −7.36769 1.46552i −0.260000 0.0517172i
\(804\) 0 0
\(805\) −9.58136 + 14.3395i −0.337699 + 0.505402i
\(806\) 0 0
\(807\) −34.2839 82.7687i −1.20685 2.91360i
\(808\) 0 0
\(809\) −3.82483 + 9.23396i −0.134474 + 0.324649i −0.976745 0.214406i \(-0.931218\pi\)
0.842271 + 0.539055i \(0.181218\pi\)
\(810\) 0 0
\(811\) 9.50273 + 47.7735i 0.333686 + 1.67755i 0.675168 + 0.737664i \(0.264071\pi\)
−0.341482 + 0.939888i \(0.610929\pi\)
\(812\) 0 0
\(813\) 3.54134 + 5.29999i 0.124200 + 0.185879i
\(814\) 0 0
\(815\) 7.32094 + 7.32094i 0.256441 + 0.256441i
\(816\) 0 0
\(817\) −7.01296 + 7.01296i −0.245352 + 0.245352i
\(818\) 0 0
\(819\) 21.4856 14.3562i 0.750768 0.501647i
\(820\) 0 0
\(821\) 13.9146 2.76779i 0.485623 0.0965965i 0.0537945 0.998552i \(-0.482868\pi\)
0.431829 + 0.901956i \(0.357868\pi\)
\(822\) 0 0
\(823\) 42.1742 + 17.4691i 1.47010 + 0.608935i 0.966882 0.255225i \(-0.0821497\pi\)
0.503217 + 0.864160i \(0.332150\pi\)
\(824\) 0 0
\(825\) −25.7608 + 10.6705i −0.896876 + 0.371498i
\(826\) 0 0
\(827\) 18.5286 + 12.3804i 0.644302 + 0.430509i 0.834326 0.551271i \(-0.185857\pi\)
−0.190025 + 0.981779i \(0.560857\pi\)
\(828\) 0 0
\(829\) −8.53766 + 42.9217i −0.296525 + 1.49073i 0.489208 + 0.872167i \(0.337286\pi\)
−0.785733 + 0.618565i \(0.787714\pi\)
\(830\) 0 0
\(831\) 14.1218 0.489881
\(832\) 0 0
\(833\) 8.09256 0.280391
\(834\) 0 0
\(835\) 0.148664 0.747383i 0.00514472 0.0258643i
\(836\) 0 0
\(837\) −8.80471 5.88312i −0.304335 0.203350i
\(838\) 0 0
\(839\) 31.4108 13.0108i 1.08442 0.449182i 0.232364 0.972629i \(-0.425354\pi\)
0.852059 + 0.523446i \(0.175354\pi\)
\(840\) 0 0
\(841\) −21.5300 8.91801i −0.742414 0.307518i
\(842\) 0 0
\(843\) 5.51560 1.09712i 0.189967 0.0377868i
\(844\) 0 0
\(845\) −18.7926 + 12.5568i −0.646487 + 0.431969i
\(846\) 0 0
\(847\) −10.1402 + 10.1402i −0.348421 + 0.348421i
\(848\) 0 0
\(849\) −19.6069 19.6069i −0.672906 0.672906i
\(850\) 0 0
\(851\) 3.65280 + 5.46680i 0.125216 + 0.187400i
\(852\) 0 0
\(853\) 10.2217 + 51.3880i 0.349984 + 1.75949i 0.608568 + 0.793502i \(0.291744\pi\)
−0.258583 + 0.965989i \(0.583256\pi\)
\(854\) 0 0
\(855\) −25.8810 + 62.4822i −0.885111 + 2.13685i
\(856\) 0 0
\(857\) 5.90193 + 14.2485i 0.201606 + 0.486720i 0.992055 0.125808i \(-0.0401524\pi\)
−0.790448 + 0.612529i \(0.790152\pi\)
\(858\) 0 0
\(859\) 32.4534 48.5700i 1.10730 1.65719i 0.484550 0.874764i \(-0.338983\pi\)
0.622747 0.782423i \(-0.286017\pi\)
\(860\) 0 0
\(861\) 72.1270 + 14.3470i 2.45808 + 0.488943i
\(862\) 0 0
\(863\) 32.8866i 1.11947i 0.828671 + 0.559736i \(0.189098\pi\)
−0.828671 + 0.559736i \(0.810902\pi\)
\(864\) 0 0
\(865\) 46.3552i 1.57612i
\(866\) 0 0
\(867\) 17.5884 + 3.49855i 0.597334 + 0.118817i
\(868\) 0 0
\(869\) 17.7512 26.5666i 0.602169 0.901209i
\(870\) 0 0
\(871\) −13.6090 32.8551i −0.461124 1.11325i
\(872\) 0 0
\(873\) 16.4444 39.7004i 0.556560 1.34365i
\(874\) 0 0
\(875\) −0.311417 1.56560i −0.0105278 0.0529269i
\(876\) 0 0
\(877\) 6.14533 + 9.19714i 0.207513 + 0.310565i 0.920597 0.390513i \(-0.127702\pi\)
−0.713084 + 0.701078i \(0.752702\pi\)
\(878\) 0 0
\(879\) 24.0520 + 24.0520i 0.811255 + 0.811255i
\(880\) 0 0
\(881\) 11.2198 11.2198i 0.378006 0.378006i −0.492376 0.870382i \(-0.663872\pi\)
0.870382 + 0.492376i \(0.163872\pi\)
\(882\) 0 0
\(883\) −29.8483 + 19.9440i −1.00447 + 0.671168i −0.945003 0.327062i \(-0.893941\pi\)
−0.0594714 + 0.998230i \(0.518941\pi\)
\(884\) 0 0
\(885\) 74.0727 14.7340i 2.48993 0.495277i
\(886\) 0 0
\(887\) 32.5314 + 13.4749i 1.09230 + 0.452444i 0.854807 0.518946i \(-0.173676\pi\)
0.237489 + 0.971390i \(0.423676\pi\)
\(888\) 0 0
\(889\) −0.594236 + 0.246141i −0.0199300 + 0.00825529i
\(890\) 0 0
\(891\) 2.43891 + 1.62963i 0.0817067 + 0.0545947i
\(892\) 0 0
\(893\) 2.68314 13.4891i 0.0897878 0.451394i
\(894\) 0 0
\(895\) −36.0167 −1.20391
\(896\) 0 0
\(897\) 17.6918 0.590713
\(898\) 0 0
\(899\) 0.843195 4.23903i 0.0281221 0.141379i
\(900\) 0 0
\(901\) −22.1388 14.7927i −0.737551 0.492816i
\(902\) 0 0
\(903\) 12.9319 5.35655i 0.430346 0.178255i
\(904\) 0 0
\(905\) −24.5358 10.1631i −0.815597 0.337831i
\(906\) 0 0
\(907\) 49.0725 9.76113i 1.62943 0.324113i 0.706095 0.708117i \(-0.250455\pi\)
0.923331 + 0.384004i \(0.125455\pi\)
\(908\) 0 0
\(909\) 62.4080 41.6997i 2.06994 1.38309i
\(910\) 0 0
\(911\) 0.673519 0.673519i 0.0223147 0.0223147i −0.695861 0.718176i \(-0.744977\pi\)
0.718176 + 0.695861i \(0.244977\pi\)
\(912\) 0 0
\(913\) 9.79066 + 9.79066i 0.324024 + 0.324024i
\(914\) 0 0
\(915\) −49.5032 74.0868i −1.63653 2.44923i
\(916\) 0 0
\(917\) −2.23468 11.2345i −0.0737955 0.370995i
\(918\) 0 0
\(919\) −18.8450 + 45.4958i −0.621639 + 1.50077i 0.228139 + 0.973628i \(0.426736\pi\)
−0.849778 + 0.527140i \(0.823264\pi\)
\(920\) 0 0
\(921\) 12.5411 + 30.2768i 0.413242 + 0.997654i
\(922\) 0 0
\(923\) −9.26185 + 13.8613i −0.304857 + 0.456251i
\(924\) 0 0
\(925\) 11.8262 + 2.35238i 0.388843 + 0.0773456i
\(926\) 0 0
\(927\) 28.4038i 0.932904i
\(928\) 0 0
\(929\) 37.0292i 1.21489i −0.794362 0.607445i \(-0.792195\pi\)
0.794362 0.607445i \(-0.207805\pi\)
\(930\) 0 0
\(931\) 10.3897 + 2.06664i 0.340510 + 0.0677316i
\(932\) 0 0
\(933\) −36.8250 + 55.1124i −1.20559 + 1.80430i
\(934\) 0 0
\(935\) −8.06323 19.4664i −0.263696 0.636618i
\(936\) 0 0
\(937\) −13.0321 + 31.4623i −0.425740 + 1.02783i 0.554884 + 0.831928i \(0.312763\pi\)
−0.980624 + 0.195900i \(0.937237\pi\)
\(938\) 0 0
\(939\) −0.995217 5.00329i −0.0324777 0.163276i
\(940\) 0 0
\(941\) −14.1954 21.2450i −0.462758 0.692566i 0.524551 0.851379i \(-0.324233\pi\)
−0.987309 + 0.158813i \(0.949233\pi\)
\(942\) 0 0
\(943\) 22.3504 + 22.3504i 0.727831 + 0.727831i
\(944\) 0 0
\(945\) 27.4749 27.4749i 0.893758 0.893758i
\(946\) 0 0
\(947\) 43.8404 29.2932i 1.42462 0.951902i 0.425730 0.904850i \(-0.360017\pi\)
0.998892 0.0470518i \(-0.0149826\pi\)
\(948\) 0 0
\(949\) 8.57321 1.70532i 0.278298 0.0553570i
\(950\) 0 0
\(951\) −39.0236 16.1641i −1.26543 0.524157i
\(952\) 0 0
\(953\) 2.18380 0.904560i 0.0707403 0.0293016i −0.347033 0.937853i \(-0.612811\pi\)
0.417773 + 0.908551i \(0.362811\pi\)
\(954\) 0 0
\(955\) −16.3960 10.9555i −0.530562 0.354510i
\(956\) 0 0
\(957\) −2.72761 + 13.7126i −0.0881711 + 0.443266i
\(958\) 0 0
\(959\) 36.6250 1.18268
\(960\) 0 0
\(961\) 27.7205 0.894210
\(962\) 0 0
\(963\) −12.7945 + 64.3221i −0.412296 + 2.07275i
\(964\) 0 0
\(965\) −27.3153 18.2515i −0.879312 0.587537i
\(966\) 0 0
\(967\) −44.4518 + 18.4125i −1.42947 + 0.592107i −0.957221 0.289357i \(-0.906559\pi\)
−0.472251 + 0.881464i \(0.656559\pi\)
\(968\) 0 0
\(969\) −36.6794 15.1931i −1.17831 0.488074i
\(970\) 0 0
\(971\) 8.20968 1.63301i 0.263461 0.0524057i −0.0615920 0.998101i \(-0.519618\pi\)
0.325053 + 0.945696i \(0.394618\pi\)
\(972\) 0 0
\(973\) −21.1059 + 14.1025i −0.676625 + 0.452106i
\(974\) 0 0
\(975\) 22.9425 22.9425i 0.734749 0.734749i
\(976\) 0 0
\(977\) 13.9697 + 13.9697i 0.446930 + 0.446930i 0.894333 0.447403i \(-0.147651\pi\)
−0.447403 + 0.894333i \(0.647651\pi\)
\(978\) 0 0
\(979\) −9.30447 13.9251i −0.297372 0.445049i
\(980\) 0 0
\(981\) −9.57607 48.1421i −0.305740 1.53706i
\(982\) 0 0
\(983\) 18.1142 43.7315i 0.577753 1.39482i −0.317072 0.948401i \(-0.602700\pi\)
0.894825 0.446417i \(-0.147300\pi\)
\(984\) 0 0
\(985\) 18.7788 + 45.3361i 0.598344 + 1.44453i
\(986\) 0 0
\(987\) −10.7839 + 16.1392i −0.343255 + 0.513718i
\(988\) 0 0
\(989\) 5.90061 + 1.17370i 0.187628 + 0.0373216i
\(990\) 0 0
\(991\) 14.8761i 0.472553i 0.971686 + 0.236277i \(0.0759272\pi\)
−0.971686 + 0.236277i \(0.924073\pi\)
\(992\) 0 0
\(993\) 15.9574i 0.506392i
\(994\) 0 0
\(995\) −84.2638 16.7611i −2.67134 0.531363i
\(996\) 0 0
\(997\) 17.6104 26.3559i 0.557728 0.834698i −0.440276 0.897863i \(-0.645119\pi\)
0.998003 + 0.0631644i \(0.0201193\pi\)
\(998\) 0 0
\(999\) −5.66878 13.6856i −0.179352 0.432994i
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 256.2.i.a.81.1 56
4.3 odd 2 64.2.i.a.61.4 yes 56
8.3 odd 2 512.2.i.b.417.1 56
8.5 even 2 512.2.i.a.417.7 56
12.11 even 2 576.2.bd.a.253.4 56
64.11 odd 16 512.2.i.b.97.1 56
64.21 even 16 inner 256.2.i.a.177.1 56
64.43 odd 16 64.2.i.a.21.4 56
64.53 even 16 512.2.i.a.97.7 56
192.107 even 16 576.2.bd.a.469.4 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
64.2.i.a.21.4 56 64.43 odd 16
64.2.i.a.61.4 yes 56 4.3 odd 2
256.2.i.a.81.1 56 1.1 even 1 trivial
256.2.i.a.177.1 56 64.21 even 16 inner
512.2.i.a.97.7 56 64.53 even 16
512.2.i.a.417.7 56 8.5 even 2
512.2.i.b.97.1 56 64.11 odd 16
512.2.i.b.417.1 56 8.3 odd 2
576.2.bd.a.253.4 56 12.11 even 2
576.2.bd.a.469.4 56 192.107 even 16