Properties

Label 392.2.y.a.305.3
Level $392$
Weight $2$
Character 392.305
Analytic conductor $3.130$
Analytic rank $0$
Dimension $84$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,2,Mod(9,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.y (of order \(21\), degree \(12\), 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: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label 305.3
Character \(\chi\) \(=\) 392.305
Dual form 392.2.y.a.9.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.51640 + 0.228560i) q^{3} +(-0.147325 + 0.375377i) q^{5} +(0.926271 - 2.47831i) q^{7} +(-0.619491 + 0.191088i) q^{9} +(-1.34370 - 0.414477i) q^{11} +(-0.0982465 - 0.430446i) q^{13} +(0.137607 - 0.602894i) q^{15} +(5.37871 - 3.66714i) q^{17} +(-4.20609 - 7.28515i) q^{19} +(-0.838153 + 3.96982i) q^{21} +(-4.38971 - 2.99285i) q^{23} +(3.54606 + 3.29026i) q^{25} +(5.04070 - 2.42747i) q^{27} +(-3.31717 - 1.59747i) q^{29} +(4.14313 - 7.17612i) q^{31} +(2.13232 + 0.321396i) q^{33} +(0.793838 + 0.712817i) q^{35} +(-0.477843 - 6.37637i) q^{37} +(0.247364 + 0.630273i) q^{39} +(0.640272 + 0.802876i) q^{41} +(-2.47117 + 3.09875i) q^{43} +(0.0195364 - 0.260695i) q^{45} +(-4.24317 + 3.93709i) q^{47} +(-5.28404 - 4.59117i) q^{49} +(-7.31810 + 6.79021i) q^{51} +(-0.822303 + 10.9729i) q^{53} +(0.353546 - 0.443332i) q^{55} +(8.04320 + 10.0859i) q^{57} +(2.38828 + 6.08524i) q^{59} +(-0.315828 - 4.21444i) q^{61} +(-0.100242 + 1.71229i) q^{63} +(0.176054 + 0.0265358i) q^{65} +(-4.22717 + 7.32168i) q^{67} +(7.34060 + 3.53505i) q^{69} +(11.6077 - 5.58995i) q^{71} +(-6.31342 - 5.85799i) q^{73} +(-6.12926 - 4.17886i) q^{75} +(-2.27184 + 2.94619i) q^{77} +(5.11546 + 8.86024i) q^{79} +(-5.48196 + 3.73754i) q^{81} +(-1.40508 + 6.15605i) q^{83} +(0.584144 + 2.55930i) q^{85} +(5.39528 + 1.66422i) q^{87} +(2.35427 - 0.726195i) q^{89} +(-1.15778 - 0.155224i) q^{91} +(-4.64247 + 11.8288i) q^{93} +(3.35434 - 0.505585i) q^{95} -0.693167 q^{97} +0.911613 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84 q - 8 q^{3} - q^{5} + 4 q^{7} + 41 q^{9} + q^{11} - 2 q^{13} + q^{15} - 5 q^{17} - 13 q^{19} + 12 q^{21} - 5 q^{23} + 2 q^{25} + 4 q^{27} - 21 q^{29} + 23 q^{31} - 17 q^{33} - 22 q^{35} - 60 q^{37}+ \cdots - 94 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{20}{21}\right)\)

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 0
\(3\) −1.51640 + 0.228560i −0.875494 + 0.131959i −0.571389 0.820679i \(-0.693595\pi\)
−0.304104 + 0.952639i \(0.598357\pi\)
\(4\) 0 0
\(5\) −0.147325 + 0.375377i −0.0658856 + 0.167874i −0.960025 0.279916i \(-0.909693\pi\)
0.894139 + 0.447790i \(0.147789\pi\)
\(6\) 0 0
\(7\) 0.926271 2.47831i 0.350098 0.936713i
\(8\) 0 0
\(9\) −0.619491 + 0.191088i −0.206497 + 0.0636959i
\(10\) 0 0
\(11\) −1.34370 0.414477i −0.405141 0.124970i 0.0854853 0.996339i \(-0.472756\pi\)
−0.490627 + 0.871370i \(0.663232\pi\)
\(12\) 0 0
\(13\) −0.0982465 0.430446i −0.0272487 0.119384i 0.959474 0.281796i \(-0.0909300\pi\)
−0.986723 + 0.162411i \(0.948073\pi\)
\(14\) 0 0
\(15\) 0.137607 0.602894i 0.0355299 0.155667i
\(16\) 0 0
\(17\) 5.37871 3.66714i 1.30453 0.889412i 0.306481 0.951877i \(-0.400848\pi\)
0.998047 + 0.0624646i \(0.0198961\pi\)
\(18\) 0 0
\(19\) −4.20609 7.28515i −0.964942 1.67133i −0.709771 0.704433i \(-0.751201\pi\)
−0.255172 0.966896i \(-0.582132\pi\)
\(20\) 0 0
\(21\) −0.838153 + 3.96982i −0.182900 + 0.866285i
\(22\) 0 0
\(23\) −4.38971 2.99285i −0.915317 0.624053i 0.0113499 0.999936i \(-0.496387\pi\)
−0.926667 + 0.375883i \(0.877340\pi\)
\(24\) 0 0
\(25\) 3.54606 + 3.29026i 0.709211 + 0.658052i
\(26\) 0 0
\(27\) 5.04070 2.42747i 0.970084 0.467168i
\(28\) 0 0
\(29\) −3.31717 1.59747i −0.615983 0.296642i 0.0997533 0.995012i \(-0.468195\pi\)
−0.715737 + 0.698370i \(0.753909\pi\)
\(30\) 0 0
\(31\) 4.14313 7.17612i 0.744129 1.28887i −0.206472 0.978452i \(-0.566198\pi\)
0.950601 0.310416i \(-0.100468\pi\)
\(32\) 0 0
\(33\) 2.13232 + 0.321396i 0.371190 + 0.0559478i
\(34\) 0 0
\(35\) 0.793838 + 0.712817i 0.134183 + 0.120488i
\(36\) 0 0
\(37\) −0.477843 6.37637i −0.0785569 1.04827i −0.887629 0.460559i \(-0.847649\pi\)
0.809072 0.587710i \(-0.199970\pi\)
\(38\) 0 0
\(39\) 0.247364 + 0.630273i 0.0396099 + 0.100924i
\(40\) 0 0
\(41\) 0.640272 + 0.802876i 0.0999937 + 0.125388i 0.829311 0.558788i \(-0.188733\pi\)
−0.729317 + 0.684176i \(0.760162\pi\)
\(42\) 0 0
\(43\) −2.47117 + 3.09875i −0.376850 + 0.472555i −0.933699 0.358059i \(-0.883439\pi\)
0.556849 + 0.830614i \(0.312010\pi\)
\(44\) 0 0
\(45\) 0.0195364 0.260695i 0.00291231 0.0388621i
\(46\) 0 0
\(47\) −4.24317 + 3.93709i −0.618930 + 0.574283i −0.926056 0.377386i \(-0.876823\pi\)
0.307126 + 0.951669i \(0.400633\pi\)
\(48\) 0 0
\(49\) −5.28404 4.59117i −0.754863 0.655882i
\(50\) 0 0
\(51\) −7.31810 + 6.79021i −1.02474 + 0.950819i
\(52\) 0 0
\(53\) −0.822303 + 10.9729i −0.112952 + 1.50724i 0.596573 + 0.802559i \(0.296529\pi\)
−0.709525 + 0.704681i \(0.751090\pi\)
\(54\) 0 0
\(55\) 0.353546 0.443332i 0.0476721 0.0597789i
\(56\) 0 0
\(57\) 8.04320 + 10.0859i 1.06535 + 1.33590i
\(58\) 0 0
\(59\) 2.38828 + 6.08524i 0.310928 + 0.792230i 0.997852 + 0.0655020i \(0.0208649\pi\)
−0.686925 + 0.726728i \(0.741040\pi\)
\(60\) 0 0
\(61\) −0.315828 4.21444i −0.0404377 0.539603i −0.980231 0.197859i \(-0.936601\pi\)
0.939793 0.341745i \(-0.111018\pi\)
\(62\) 0 0
\(63\) −0.100242 + 1.71229i −0.0126293 + 0.215728i
\(64\) 0 0
\(65\) 0.176054 + 0.0265358i 0.0218368 + 0.00329136i
\(66\) 0 0
\(67\) −4.22717 + 7.32168i −0.516431 + 0.894485i 0.483387 + 0.875407i \(0.339407\pi\)
−0.999818 + 0.0190783i \(0.993927\pi\)
\(68\) 0 0
\(69\) 7.34060 + 3.53505i 0.883704 + 0.425569i
\(70\) 0 0
\(71\) 11.6077 5.58995i 1.37757 0.663405i 0.409094 0.912492i \(-0.365845\pi\)
0.968481 + 0.249087i \(0.0801307\pi\)
\(72\) 0 0
\(73\) −6.31342 5.85799i −0.738929 0.685626i 0.217999 0.975949i \(-0.430047\pi\)
−0.956929 + 0.290323i \(0.906237\pi\)
\(74\) 0 0
\(75\) −6.12926 4.17886i −0.707746 0.482533i
\(76\) 0 0
\(77\) −2.27184 + 2.94619i −0.258900 + 0.335750i
\(78\) 0 0
\(79\) 5.11546 + 8.86024i 0.575535 + 0.996855i 0.995983 + 0.0895390i \(0.0285394\pi\)
−0.420449 + 0.907316i \(0.638127\pi\)
\(80\) 0 0
\(81\) −5.48196 + 3.73754i −0.609107 + 0.415282i
\(82\) 0 0
\(83\) −1.40508 + 6.15605i −0.154227 + 0.675715i 0.837401 + 0.546589i \(0.184074\pi\)
−0.991628 + 0.129125i \(0.958783\pi\)
\(84\) 0 0
\(85\) 0.584144 + 2.55930i 0.0633593 + 0.277595i
\(86\) 0 0
\(87\) 5.39528 + 1.66422i 0.578434 + 0.178423i
\(88\) 0 0
\(89\) 2.35427 0.726195i 0.249552 0.0769765i −0.167456 0.985880i \(-0.553555\pi\)
0.417007 + 0.908903i \(0.363079\pi\)
\(90\) 0 0
\(91\) −1.15778 0.155224i −0.121369 0.0162719i
\(92\) 0 0
\(93\) −4.64247 + 11.8288i −0.481402 + 1.22659i
\(94\) 0 0
\(95\) 3.35434 0.505585i 0.344148 0.0518719i
\(96\) 0 0
\(97\) −0.693167 −0.0703805 −0.0351902 0.999381i \(-0.511204\pi\)
−0.0351902 + 0.999381i \(0.511204\pi\)
\(98\) 0 0
\(99\) 0.911613 0.0916206
\(100\) 0 0
\(101\) 7.19439 1.08438i 0.715868 0.107900i 0.218999 0.975725i \(-0.429721\pi\)
0.496869 + 0.867825i \(0.334483\pi\)
\(102\) 0 0
\(103\) 2.15610 5.49366i 0.212447 0.541306i −0.784435 0.620211i \(-0.787047\pi\)
0.996882 + 0.0789048i \(0.0251423\pi\)
\(104\) 0 0
\(105\) −1.36670 0.899475i −0.133376 0.0877798i
\(106\) 0 0
\(107\) −10.1482 + 3.13032i −0.981068 + 0.302619i −0.743488 0.668749i \(-0.766830\pi\)
−0.237579 + 0.971368i \(0.576354\pi\)
\(108\) 0 0
\(109\) 5.04479 + 1.55611i 0.483203 + 0.149049i 0.526780 0.850002i \(-0.323399\pi\)
−0.0435767 + 0.999050i \(0.513875\pi\)
\(110\) 0 0
\(111\) 2.18199 + 9.55991i 0.207105 + 0.907386i
\(112\) 0 0
\(113\) 1.30238 5.70612i 0.122518 0.536787i −0.875997 0.482316i \(-0.839796\pi\)
0.998515 0.0544707i \(-0.0173472\pi\)
\(114\) 0 0
\(115\) 1.77016 1.20687i 0.165068 0.112542i
\(116\) 0 0
\(117\) 0.143116 + 0.247884i 0.0132311 + 0.0229169i
\(118\) 0 0
\(119\) −4.10617 16.7269i −0.376412 1.53335i
\(120\) 0 0
\(121\) −7.45488 5.08265i −0.677717 0.462059i
\(122\) 0 0
\(123\) −1.15441 1.07114i −0.104090 0.0965814i
\(124\) 0 0
\(125\) −3.57410 + 1.72119i −0.319677 + 0.153948i
\(126\) 0 0
\(127\) −1.10520 0.532238i −0.0980709 0.0472285i 0.384205 0.923248i \(-0.374476\pi\)
−0.482276 + 0.876019i \(0.660190\pi\)
\(128\) 0 0
\(129\) 3.03903 5.26376i 0.267572 0.463448i
\(130\) 0 0
\(131\) −3.26855 0.492655i −0.285575 0.0430435i 0.00469285 0.999989i \(-0.498506\pi\)
−0.290268 + 0.956946i \(0.593744\pi\)
\(132\) 0 0
\(133\) −21.9508 + 3.67596i −1.90338 + 0.318746i
\(134\) 0 0
\(135\) 0.168598 + 2.24979i 0.0145106 + 0.193631i
\(136\) 0 0
\(137\) 4.27559 + 10.8940i 0.365288 + 0.930739i 0.988843 + 0.148959i \(0.0475924\pi\)
−0.623555 + 0.781779i \(0.714312\pi\)
\(138\) 0 0
\(139\) 6.76258 + 8.48001i 0.573595 + 0.719265i 0.981006 0.193980i \(-0.0621396\pi\)
−0.407411 + 0.913245i \(0.633568\pi\)
\(140\) 0 0
\(141\) 5.53448 6.94002i 0.466087 0.584455i
\(142\) 0 0
\(143\) −0.0463960 + 0.619112i −0.00387983 + 0.0517728i
\(144\) 0 0
\(145\) 1.08835 1.00984i 0.0903828 0.0838630i
\(146\) 0 0
\(147\) 9.06208 + 5.75433i 0.747428 + 0.474609i
\(148\) 0 0
\(149\) 13.4472 12.4771i 1.10163 1.02217i 0.102010 0.994783i \(-0.467473\pi\)
0.999625 0.0273845i \(-0.00871783\pi\)
\(150\) 0 0
\(151\) 0.866609 11.5641i 0.0705236 0.941072i −0.843949 0.536424i \(-0.819775\pi\)
0.914472 0.404649i \(-0.132606\pi\)
\(152\) 0 0
\(153\) −2.63132 + 3.29957i −0.212729 + 0.266754i
\(154\) 0 0
\(155\) 2.08336 + 2.61246i 0.167340 + 0.209837i
\(156\) 0 0
\(157\) −7.47975 19.0581i −0.596949 1.52100i −0.835284 0.549818i \(-0.814697\pi\)
0.238336 0.971183i \(-0.423398\pi\)
\(158\) 0 0
\(159\) −1.26102 16.8272i −0.100006 1.33448i
\(160\) 0 0
\(161\) −11.4833 + 8.10686i −0.905009 + 0.638910i
\(162\) 0 0
\(163\) −2.52434 0.380483i −0.197721 0.0298017i 0.0494349 0.998777i \(-0.484258\pi\)
−0.247156 + 0.968976i \(0.579496\pi\)
\(164\) 0 0
\(165\) −0.434788 + 0.753075i −0.0338482 + 0.0586268i
\(166\) 0 0
\(167\) 11.2810 + 5.43264i 0.872949 + 0.420390i 0.816044 0.577989i \(-0.196163\pi\)
0.0569050 + 0.998380i \(0.481877\pi\)
\(168\) 0 0
\(169\) 11.5370 5.55591i 0.887459 0.427378i
\(170\) 0 0
\(171\) 3.99774 + 3.70936i 0.305715 + 0.283662i
\(172\) 0 0
\(173\) 6.65833 + 4.53957i 0.506224 + 0.345137i 0.789348 0.613946i \(-0.210419\pi\)
−0.283125 + 0.959083i \(0.591371\pi\)
\(174\) 0 0
\(175\) 11.4389 5.74055i 0.864699 0.433945i
\(176\) 0 0
\(177\) −5.01243 8.68178i −0.376757 0.652563i
\(178\) 0 0
\(179\) −4.13472 + 2.81900i −0.309043 + 0.210702i −0.707903 0.706310i \(-0.750358\pi\)
0.398859 + 0.917012i \(0.369406\pi\)
\(180\) 0 0
\(181\) 3.24591 14.2213i 0.241267 1.05706i −0.698599 0.715513i \(-0.746193\pi\)
0.939866 0.341545i \(-0.110950\pi\)
\(182\) 0 0
\(183\) 1.44218 + 6.31858i 0.106609 + 0.467083i
\(184\) 0 0
\(185\) 2.46394 + 0.760025i 0.181153 + 0.0558781i
\(186\) 0 0
\(187\) −8.74733 + 2.69819i −0.639668 + 0.197311i
\(188\) 0 0
\(189\) −1.34698 14.7409i −0.0979782 1.07224i
\(190\) 0 0
\(191\) −7.83441 + 19.9618i −0.566878 + 1.44438i 0.303899 + 0.952704i \(0.401711\pi\)
−0.870777 + 0.491678i \(0.836384\pi\)
\(192\) 0 0
\(193\) −20.2138 + 3.04674i −1.45502 + 0.219309i −0.828454 0.560057i \(-0.810779\pi\)
−0.626569 + 0.779366i \(0.715541\pi\)
\(194\) 0 0
\(195\) −0.273033 −0.0195523
\(196\) 0 0
\(197\) 5.14452 0.366532 0.183266 0.983063i \(-0.441333\pi\)
0.183266 + 0.983063i \(0.441333\pi\)
\(198\) 0 0
\(199\) −5.16760 + 0.778891i −0.366322 + 0.0552141i −0.329624 0.944112i \(-0.606922\pi\)
−0.0366978 + 0.999326i \(0.511684\pi\)
\(200\) 0 0
\(201\) 4.73664 12.0688i 0.334097 0.851264i
\(202\) 0 0
\(203\) −7.03162 + 6.74129i −0.493523 + 0.473146i
\(204\) 0 0
\(205\) −0.395709 + 0.122060i −0.0276375 + 0.00852504i
\(206\) 0 0
\(207\) 3.29128 + 1.01523i 0.228760 + 0.0705631i
\(208\) 0 0
\(209\) 2.63220 + 11.5324i 0.182073 + 0.797713i
\(210\) 0 0
\(211\) 4.91333 21.5267i 0.338248 1.48196i −0.464464 0.885592i \(-0.653753\pi\)
0.802712 0.596367i \(-0.203390\pi\)
\(212\) 0 0
\(213\) −16.3242 + 11.1296i −1.11852 + 0.762591i
\(214\) 0 0
\(215\) −0.799135 1.38414i −0.0545006 0.0943978i
\(216\) 0 0
\(217\) −13.9470 16.9150i −0.946783 1.14827i
\(218\) 0 0
\(219\) 10.9126 + 7.44006i 0.737403 + 0.502753i
\(220\) 0 0
\(221\) −2.10695 1.95496i −0.141728 0.131505i
\(222\) 0 0
\(223\) 1.74693 0.841275i 0.116983 0.0563359i −0.374477 0.927236i \(-0.622178\pi\)
0.491460 + 0.870900i \(0.336463\pi\)
\(224\) 0 0
\(225\) −2.82548 1.36068i −0.188365 0.0907119i
\(226\) 0 0
\(227\) 6.02443 10.4346i 0.399855 0.692570i −0.593852 0.804574i \(-0.702394\pi\)
0.993708 + 0.112004i \(0.0357270\pi\)
\(228\) 0 0
\(229\) 14.2131 + 2.14228i 0.939227 + 0.141566i 0.600775 0.799418i \(-0.294859\pi\)
0.338452 + 0.940984i \(0.390097\pi\)
\(230\) 0 0
\(231\) 2.77163 4.98686i 0.182360 0.328111i
\(232\) 0 0
\(233\) 1.55243 + 20.7158i 0.101703 + 1.35713i 0.781578 + 0.623807i \(0.214415\pi\)
−0.679875 + 0.733328i \(0.737966\pi\)
\(234\) 0 0
\(235\) −0.852768 2.17282i −0.0556285 0.141739i
\(236\) 0 0
\(237\) −9.78219 12.2665i −0.635421 0.796793i
\(238\) 0 0
\(239\) 12.4710 15.6382i 0.806684 1.01155i −0.192856 0.981227i \(-0.561775\pi\)
0.999540 0.0303225i \(-0.00965343\pi\)
\(240\) 0 0
\(241\) −0.308807 + 4.12075i −0.0198920 + 0.265441i 0.978386 + 0.206786i \(0.0663002\pi\)
−0.998278 + 0.0586550i \(0.981319\pi\)
\(242\) 0 0
\(243\) −4.84516 + 4.49565i −0.310817 + 0.288396i
\(244\) 0 0
\(245\) 2.50189 1.30712i 0.159840 0.0835085i
\(246\) 0 0
\(247\) −2.72263 + 2.52623i −0.173237 + 0.160740i
\(248\) 0 0
\(249\) 0.723631 9.65618i 0.0458582 0.611936i
\(250\) 0 0
\(251\) −12.1121 + 15.1881i −0.764509 + 0.958664i −0.999912 0.0132335i \(-0.995788\pi\)
0.235403 + 0.971898i \(0.424359\pi\)
\(252\) 0 0
\(253\) 4.65799 + 5.84094i 0.292845 + 0.367216i
\(254\) 0 0
\(255\) −1.47075 3.74741i −0.0921020 0.234672i
\(256\) 0 0
\(257\) −1.93171 25.7769i −0.120497 1.60792i −0.650217 0.759749i \(-0.725322\pi\)
0.529720 0.848173i \(-0.322297\pi\)
\(258\) 0 0
\(259\) −16.2452 4.72201i −1.00943 0.293411i
\(260\) 0 0
\(261\) 2.36021 + 0.355745i 0.146094 + 0.0220201i
\(262\) 0 0
\(263\) 10.0884 17.4736i 0.622076 1.07747i −0.367022 0.930212i \(-0.619623\pi\)
0.989098 0.147256i \(-0.0470440\pi\)
\(264\) 0 0
\(265\) −3.99782 1.92525i −0.245584 0.118267i
\(266\) 0 0
\(267\) −3.40403 + 1.63929i −0.208323 + 0.100323i
\(268\) 0 0
\(269\) −20.0231 18.5787i −1.22083 1.13276i −0.987040 0.160476i \(-0.948697\pi\)
−0.233789 0.972287i \(-0.575113\pi\)
\(270\) 0 0
\(271\) 25.5947 + 17.4502i 1.55477 + 1.06002i 0.968217 + 0.250113i \(0.0804678\pi\)
0.586553 + 0.809911i \(0.300485\pi\)
\(272\) 0 0
\(273\) 1.79114 0.0292408i 0.108405 0.00176974i
\(274\) 0 0
\(275\) −3.40111 5.89089i −0.205094 0.355234i
\(276\) 0 0
\(277\) 11.3230 7.71991i 0.680335 0.463844i −0.173196 0.984887i \(-0.555409\pi\)
0.853530 + 0.521043i \(0.174457\pi\)
\(278\) 0 0
\(279\) −1.19537 + 5.23724i −0.0715647 + 0.313546i
\(280\) 0 0
\(281\) 6.90340 + 30.2458i 0.411822 + 1.80431i 0.575505 + 0.817798i \(0.304806\pi\)
−0.163683 + 0.986513i \(0.552337\pi\)
\(282\) 0 0
\(283\) 30.2330 + 9.32565i 1.79717 + 0.554353i 0.998853 0.0478883i \(-0.0152492\pi\)
0.798314 + 0.602241i \(0.205725\pi\)
\(284\) 0 0
\(285\) −4.97096 + 1.53334i −0.294454 + 0.0908271i
\(286\) 0 0
\(287\) 2.58284 0.843112i 0.152460 0.0497674i
\(288\) 0 0
\(289\) 9.27177 23.6241i 0.545398 1.38965i
\(290\) 0 0
\(291\) 1.05112 0.158431i 0.0616176 0.00928737i
\(292\) 0 0
\(293\) 4.71694 0.275566 0.137783 0.990462i \(-0.456002\pi\)
0.137783 + 0.990462i \(0.456002\pi\)
\(294\) 0 0
\(295\) −2.63611 −0.153480
\(296\) 0 0
\(297\) −7.77934 + 1.17255i −0.451403 + 0.0680380i
\(298\) 0 0
\(299\) −0.856988 + 2.18357i −0.0495609 + 0.126279i
\(300\) 0 0
\(301\) 5.39069 + 8.99461i 0.310714 + 0.518441i
\(302\) 0 0
\(303\) −10.6617 + 3.28870i −0.612500 + 0.188931i
\(304\) 0 0
\(305\) 1.62853 + 0.502336i 0.0932495 + 0.0287637i
\(306\) 0 0
\(307\) 1.59463 + 6.98652i 0.0910102 + 0.398742i 0.999830 0.0184476i \(-0.00587238\pi\)
−0.908820 + 0.417189i \(0.863015\pi\)
\(308\) 0 0
\(309\) −2.01388 + 8.82338i −0.114566 + 0.501944i
\(310\) 0 0
\(311\) −1.42254 + 0.969874i −0.0806651 + 0.0549965i −0.602981 0.797756i \(-0.706021\pi\)
0.522316 + 0.852752i \(0.325068\pi\)
\(312\) 0 0
\(313\) 4.82195 + 8.35187i 0.272553 + 0.472076i 0.969515 0.245033i \(-0.0787987\pi\)
−0.696962 + 0.717108i \(0.745465\pi\)
\(314\) 0 0
\(315\) −0.627986 0.289891i −0.0353830 0.0163335i
\(316\) 0 0
\(317\) 6.68310 + 4.55646i 0.375361 + 0.255916i 0.736275 0.676682i \(-0.236583\pi\)
−0.360914 + 0.932599i \(0.617535\pi\)
\(318\) 0 0
\(319\) 3.79518 + 3.52141i 0.212489 + 0.197161i
\(320\) 0 0
\(321\) 14.6733 7.06630i 0.818985 0.394402i
\(322\) 0 0
\(323\) −49.3390 23.7604i −2.74529 1.32206i
\(324\) 0 0
\(325\) 1.06789 1.84964i 0.0592360 0.102600i
\(326\) 0 0
\(327\) −8.00559 1.20665i −0.442710 0.0667278i
\(328\) 0 0
\(329\) 5.82700 + 14.1627i 0.321253 + 0.780815i
\(330\) 0 0
\(331\) −2.06049 27.4953i −0.113255 1.51128i −0.707339 0.706875i \(-0.750105\pi\)
0.594084 0.804403i \(-0.297515\pi\)
\(332\) 0 0
\(333\) 1.51447 + 3.85880i 0.0829922 + 0.211461i
\(334\) 0 0
\(335\) −2.12562 2.66545i −0.116135 0.145629i
\(336\) 0 0
\(337\) −6.36702 + 7.98399i −0.346834 + 0.434916i −0.924398 0.381429i \(-0.875432\pi\)
0.577564 + 0.816345i \(0.304003\pi\)
\(338\) 0 0
\(339\) −0.670742 + 8.95043i −0.0364297 + 0.486121i
\(340\) 0 0
\(341\) −8.54147 + 7.92533i −0.462547 + 0.429181i
\(342\) 0 0
\(343\) −16.2728 + 8.84283i −0.878649 + 0.477468i
\(344\) 0 0
\(345\) −2.40842 + 2.23469i −0.129665 + 0.120312i
\(346\) 0 0
\(347\) −1.15304 + 15.3863i −0.0618985 + 0.825978i 0.876716 + 0.481009i \(0.159730\pi\)
−0.938614 + 0.344969i \(0.887890\pi\)
\(348\) 0 0
\(349\) 8.19211 10.2726i 0.438513 0.549878i −0.512637 0.858605i \(-0.671331\pi\)
0.951151 + 0.308727i \(0.0999029\pi\)
\(350\) 0 0
\(351\) −1.54013 1.93126i −0.0822060 0.103083i
\(352\) 0 0
\(353\) 9.60760 + 24.4798i 0.511361 + 1.30293i 0.920603 + 0.390501i \(0.127698\pi\)
−0.409241 + 0.912426i \(0.634207\pi\)
\(354\) 0 0
\(355\) 0.388246 + 5.18078i 0.0206060 + 0.274967i
\(356\) 0 0
\(357\) 10.0497 + 24.4261i 0.531886 + 1.29277i
\(358\) 0 0
\(359\) −9.24524 1.39350i −0.487945 0.0735459i −0.0995381 0.995034i \(-0.531737\pi\)
−0.388407 + 0.921488i \(0.626975\pi\)
\(360\) 0 0
\(361\) −25.8823 + 44.8295i −1.36223 + 2.35945i
\(362\) 0 0
\(363\) 12.4663 + 6.00344i 0.654310 + 0.315099i
\(364\) 0 0
\(365\) 3.12908 1.50688i 0.163783 0.0788739i
\(366\) 0 0
\(367\) −3.23045 2.99742i −0.168628 0.156464i 0.591345 0.806419i \(-0.298597\pi\)
−0.759973 + 0.649955i \(0.774788\pi\)
\(368\) 0 0
\(369\) −0.550063 0.375026i −0.0286351 0.0195231i
\(370\) 0 0
\(371\) 26.4325 + 12.2018i 1.37231 + 0.633485i
\(372\) 0 0
\(373\) 9.86013 + 17.0782i 0.510538 + 0.884278i 0.999925 + 0.0122114i \(0.00388712\pi\)
−0.489387 + 0.872067i \(0.662780\pi\)
\(374\) 0 0
\(375\) 5.02636 3.42692i 0.259560 0.176965i
\(376\) 0 0
\(377\) −0.361722 + 1.58481i −0.0186296 + 0.0816218i
\(378\) 0 0
\(379\) 4.31301 + 18.8965i 0.221544 + 0.970649i 0.956316 + 0.292335i \(0.0944321\pi\)
−0.734772 + 0.678314i \(0.762711\pi\)
\(380\) 0 0
\(381\) 1.79758 + 0.554479i 0.0920927 + 0.0284068i
\(382\) 0 0
\(383\) −19.2310 + 5.93197i −0.982657 + 0.303109i −0.744136 0.668028i \(-0.767139\pi\)
−0.238521 + 0.971137i \(0.576662\pi\)
\(384\) 0 0
\(385\) −0.771235 1.28684i −0.0393058 0.0655835i
\(386\) 0 0
\(387\) 0.938736 2.39186i 0.0477186 0.121585i
\(388\) 0 0
\(389\) 18.3243 2.76195i 0.929080 0.140036i 0.332969 0.942938i \(-0.391950\pi\)
0.596111 + 0.802902i \(0.296712\pi\)
\(390\) 0 0
\(391\) −34.5862 −1.74910
\(392\) 0 0
\(393\) 5.06903 0.255699
\(394\) 0 0
\(395\) −4.07956 + 0.614895i −0.205265 + 0.0309387i
\(396\) 0 0
\(397\) −7.20097 + 18.3478i −0.361406 + 0.920848i 0.628331 + 0.777946i \(0.283738\pi\)
−0.989737 + 0.142902i \(0.954357\pi\)
\(398\) 0 0
\(399\) 32.4461 10.5913i 1.62434 0.530229i
\(400\) 0 0
\(401\) −4.34786 + 1.34114i −0.217122 + 0.0669732i −0.401408 0.915899i \(-0.631479\pi\)
0.184286 + 0.982873i \(0.441003\pi\)
\(402\) 0 0
\(403\) −3.49598 1.07837i −0.174147 0.0537173i
\(404\) 0 0
\(405\) −0.595358 2.60843i −0.0295836 0.129614i
\(406\) 0 0
\(407\) −2.00078 + 8.76600i −0.0991751 + 0.434514i
\(408\) 0 0
\(409\) 7.22445 4.92555i 0.357226 0.243553i −0.371387 0.928478i \(-0.621118\pi\)
0.728613 + 0.684926i \(0.240165\pi\)
\(410\) 0 0
\(411\) −8.97344 15.5424i −0.442627 0.766652i
\(412\) 0 0
\(413\) 17.2933 0.282318i 0.850948 0.0138920i
\(414\) 0 0
\(415\) −2.10384 1.43437i −0.103273 0.0704106i
\(416\) 0 0
\(417\) −12.1930 11.3134i −0.597092 0.554021i
\(418\) 0 0
\(419\) 14.7585 7.10731i 0.720999 0.347215i −0.0371378 0.999310i \(-0.511824\pi\)
0.758137 + 0.652095i \(0.226110\pi\)
\(420\) 0 0
\(421\) −10.2007 4.91239i −0.497150 0.239415i 0.168468 0.985707i \(-0.446118\pi\)
−0.665619 + 0.746292i \(0.731832\pi\)
\(422\) 0 0
\(423\) 1.87628 3.24981i 0.0912277 0.158011i
\(424\) 0 0
\(425\) 31.1390 + 4.69345i 1.51047 + 0.227666i
\(426\) 0 0
\(427\) −10.7372 3.12099i −0.519611 0.151035i
\(428\) 0 0
\(429\) −0.0711496 0.949426i −0.00343514 0.0458387i
\(430\) 0 0
\(431\) −8.18877 20.8646i −0.394439 1.00501i −0.980809 0.194971i \(-0.937539\pi\)
0.586370 0.810044i \(-0.300557\pi\)
\(432\) 0 0
\(433\) −17.1243 21.4732i −0.822942 1.03194i −0.998870 0.0475334i \(-0.984864\pi\)
0.175928 0.984403i \(-0.443707\pi\)
\(434\) 0 0
\(435\) −1.41957 + 1.78008i −0.0680630 + 0.0853483i
\(436\) 0 0
\(437\) −3.33990 + 44.5679i −0.159769 + 2.13197i
\(438\) 0 0
\(439\) −15.8635 + 14.7192i −0.757123 + 0.702507i −0.961018 0.276485i \(-0.910830\pi\)
0.203895 + 0.978993i \(0.434640\pi\)
\(440\) 0 0
\(441\) 4.15074 + 1.83448i 0.197654 + 0.0873560i
\(442\) 0 0
\(443\) 0.728271 0.675737i 0.0346012 0.0321052i −0.662683 0.748900i \(-0.730582\pi\)
0.697284 + 0.716795i \(0.254392\pi\)
\(444\) 0 0
\(445\) −0.0742445 + 0.990724i −0.00351953 + 0.0469648i
\(446\) 0 0
\(447\) −17.5395 + 21.9938i −0.829590 + 1.04027i
\(448\) 0 0
\(449\) 11.1079 + 13.9289i 0.524214 + 0.657343i 0.971498 0.237049i \(-0.0761801\pi\)
−0.447284 + 0.894392i \(0.647609\pi\)
\(450\) 0 0
\(451\) −0.527561 1.34420i −0.0248419 0.0632961i
\(452\) 0 0
\(453\) 1.32897 + 17.7339i 0.0624404 + 0.833209i
\(454\) 0 0
\(455\) 0.228837 0.411736i 0.0107281 0.0193025i
\(456\) 0 0
\(457\) 28.7043 + 4.32648i 1.34273 + 0.202384i 0.780790 0.624793i \(-0.214817\pi\)
0.561941 + 0.827177i \(0.310055\pi\)
\(458\) 0 0
\(459\) 18.2106 31.5416i 0.849997 1.47224i
\(460\) 0 0
\(461\) −18.8904 9.09712i −0.879812 0.423695i −0.0612561 0.998122i \(-0.519511\pi\)
−0.818556 + 0.574427i \(0.805225\pi\)
\(462\) 0 0
\(463\) −2.61473 + 1.25919i −0.121517 + 0.0585194i −0.493654 0.869658i \(-0.664339\pi\)
0.372138 + 0.928178i \(0.378625\pi\)
\(464\) 0 0
\(465\) −3.75631 3.48535i −0.174195 0.161629i
\(466\) 0 0
\(467\) −8.56164 5.83723i −0.396186 0.270115i 0.348798 0.937198i \(-0.386590\pi\)
−0.744983 + 0.667083i \(0.767542\pi\)
\(468\) 0 0
\(469\) 14.2299 + 17.2581i 0.657075 + 0.796905i
\(470\) 0 0
\(471\) 15.6982 + 27.1901i 0.723335 + 1.25285i
\(472\) 0 0
\(473\) 4.60488 3.13955i 0.211733 0.144357i
\(474\) 0 0
\(475\) 9.05503 39.6727i 0.415473 1.82031i
\(476\) 0 0
\(477\) −1.58737 6.95473i −0.0726808 0.318435i
\(478\) 0 0
\(479\) 7.13189 + 2.19990i 0.325864 + 0.100516i 0.453371 0.891322i \(-0.350221\pi\)
−0.127507 + 0.991838i \(0.540697\pi\)
\(480\) 0 0
\(481\) −2.69774 + 0.832142i −0.123006 + 0.0379424i
\(482\) 0 0
\(483\) 15.5603 14.9179i 0.708019 0.678786i
\(484\) 0 0
\(485\) 0.102121 0.260199i 0.00463706 0.0118150i
\(486\) 0 0
\(487\) 15.7127 2.36831i 0.712011 0.107318i 0.216957 0.976181i \(-0.430387\pi\)
0.495053 + 0.868863i \(0.335149\pi\)
\(488\) 0 0
\(489\) 3.91487 0.177036
\(490\) 0 0
\(491\) −38.1651 −1.72237 −0.861184 0.508293i \(-0.830277\pi\)
−0.861184 + 0.508293i \(0.830277\pi\)
\(492\) 0 0
\(493\) −23.7002 + 3.57223i −1.06740 + 0.160885i
\(494\) 0 0
\(495\) −0.134303 + 0.342199i −0.00603647 + 0.0153807i
\(496\) 0 0
\(497\) −3.10180 33.9452i −0.139135 1.52265i
\(498\) 0 0
\(499\) 25.7258 7.93537i 1.15165 0.355236i 0.340595 0.940210i \(-0.389371\pi\)
0.811052 + 0.584974i \(0.198895\pi\)
\(500\) 0 0
\(501\) −18.3482 5.65966i −0.819736 0.252855i
\(502\) 0 0
\(503\) −4.07027 17.8330i −0.181484 0.795135i −0.980924 0.194389i \(-0.937728\pi\)
0.799440 0.600746i \(-0.205130\pi\)
\(504\) 0 0
\(505\) −0.652859 + 2.86036i −0.0290519 + 0.127284i
\(506\) 0 0
\(507\) −16.2248 + 11.0619i −0.720568 + 0.491275i
\(508\) 0 0
\(509\) −4.48061 7.76064i −0.198599 0.343984i 0.749475 0.662032i \(-0.230306\pi\)
−0.948075 + 0.318048i \(0.896973\pi\)
\(510\) 0 0
\(511\) −20.3659 + 10.2205i −0.900933 + 0.452129i
\(512\) 0 0
\(513\) −38.8861 26.5121i −1.71687 1.17054i
\(514\) 0 0
\(515\) 1.74454 + 1.61870i 0.0768738 + 0.0713285i
\(516\) 0 0
\(517\) 7.33339 3.53157i 0.322522 0.155318i
\(518\) 0 0
\(519\) −11.1343 5.36198i −0.488740 0.235365i
\(520\) 0 0
\(521\) 21.6079 37.4259i 0.946658 1.63966i 0.194262 0.980950i \(-0.437769\pi\)
0.752397 0.658710i \(-0.228898\pi\)
\(522\) 0 0
\(523\) 9.33257 + 1.40666i 0.408085 + 0.0615089i 0.349877 0.936796i \(-0.386223\pi\)
0.0582078 + 0.998304i \(0.481461\pi\)
\(524\) 0 0
\(525\) −16.0339 + 11.3195i −0.699775 + 0.494021i
\(526\) 0 0
\(527\) −4.03113 53.7917i −0.175599 2.34320i
\(528\) 0 0
\(529\) 1.90952 + 4.86539i 0.0830228 + 0.211539i
\(530\) 0 0
\(531\) −2.64233 3.31338i −0.114667 0.143788i
\(532\) 0 0
\(533\) 0.282690 0.354482i 0.0122447 0.0153543i
\(534\) 0 0
\(535\) 0.320036 4.27059i 0.0138364 0.184634i
\(536\) 0 0
\(537\) 5.62557 5.21977i 0.242761 0.225250i
\(538\) 0 0
\(539\) 5.19724 + 8.35929i 0.223861 + 0.360060i
\(540\) 0 0
\(541\) 29.2102 27.1031i 1.25585 1.16526i 0.276953 0.960883i \(-0.410675\pi\)
0.978893 0.204372i \(-0.0655152\pi\)
\(542\) 0 0
\(543\) −1.67168 + 22.3070i −0.0717386 + 0.957285i
\(544\) 0 0
\(545\) −1.32735 + 1.66444i −0.0568574 + 0.0712970i
\(546\) 0 0
\(547\) 18.5543 + 23.2664i 0.793326 + 0.994799i 0.999866 + 0.0163609i \(0.00520808\pi\)
−0.206540 + 0.978438i \(0.566220\pi\)
\(548\) 0 0
\(549\) 1.00098 + 2.55046i 0.0427208 + 0.108851i
\(550\) 0 0
\(551\) 2.31452 + 30.8852i 0.0986021 + 1.31575i
\(552\) 0 0
\(553\) 26.6967 4.47072i 1.13526 0.190114i
\(554\) 0 0
\(555\) −3.91003 0.589342i −0.165972 0.0250162i
\(556\) 0 0
\(557\) 15.7627 27.3017i 0.667885 1.15681i −0.310609 0.950538i \(-0.600533\pi\)
0.978494 0.206273i \(-0.0661335\pi\)
\(558\) 0 0
\(559\) 1.57663 + 0.759265i 0.0666843 + 0.0321135i
\(560\) 0 0
\(561\) 12.6477 6.09083i 0.533988 0.257155i
\(562\) 0 0
\(563\) 3.43099 + 3.18349i 0.144599 + 0.134168i 0.749156 0.662394i \(-0.230459\pi\)
−0.604557 + 0.796562i \(0.706650\pi\)
\(564\) 0 0
\(565\) 1.95007 + 1.32954i 0.0820402 + 0.0559341i
\(566\) 0 0
\(567\) 4.18499 + 17.0480i 0.175753 + 0.715947i
\(568\) 0 0
\(569\) −11.5923 20.0784i −0.485973 0.841730i 0.513897 0.857852i \(-0.328201\pi\)
−0.999870 + 0.0161216i \(0.994868\pi\)
\(570\) 0 0
\(571\) −7.89763 + 5.38451i −0.330505 + 0.225335i −0.717191 0.696877i \(-0.754572\pi\)
0.386685 + 0.922212i \(0.373620\pi\)
\(572\) 0 0
\(573\) 7.31763 32.0606i 0.305698 1.33935i
\(574\) 0 0
\(575\) −5.71889 25.0561i −0.238494 1.04491i
\(576\) 0 0
\(577\) −6.49392 2.00311i −0.270345 0.0833905i 0.156618 0.987659i \(-0.449941\pi\)
−0.426964 + 0.904269i \(0.640417\pi\)
\(578\) 0 0
\(579\) 29.9559 9.24016i 1.24492 0.384008i
\(580\) 0 0
\(581\) 13.9551 + 9.18440i 0.578956 + 0.381033i
\(582\) 0 0
\(583\) 5.65294 14.4034i 0.234121 0.596530i
\(584\) 0 0
\(585\) −0.114134 + 0.0172030i −0.00471887 + 0.000711256i
\(586\) 0 0
\(587\) −15.0604 −0.621609 −0.310804 0.950474i \(-0.600598\pi\)
−0.310804 + 0.950474i \(0.600598\pi\)
\(588\) 0 0
\(589\) −69.7055 −2.87216
\(590\) 0 0
\(591\) −7.80115 + 1.17583i −0.320897 + 0.0483674i
\(592\) 0 0
\(593\) −3.42593 + 8.72912i −0.140686 + 0.358462i −0.983799 0.179274i \(-0.942625\pi\)
0.843113 + 0.537736i \(0.180720\pi\)
\(594\) 0 0
\(595\) 6.88382 + 0.922918i 0.282209 + 0.0378359i
\(596\) 0 0
\(597\) 7.65813 2.36222i 0.313426 0.0966792i
\(598\) 0 0
\(599\) −8.50443 2.62327i −0.347482 0.107184i 0.116101 0.993237i \(-0.462960\pi\)
−0.463583 + 0.886053i \(0.653436\pi\)
\(600\) 0 0
\(601\) 5.92646 + 25.9655i 0.241745 + 1.05916i 0.939427 + 0.342748i \(0.111358\pi\)
−0.697682 + 0.716407i \(0.745785\pi\)
\(602\) 0 0
\(603\) 1.21961 5.34348i 0.0496665 0.217603i
\(604\) 0 0
\(605\) 3.00620 2.04959i 0.122219 0.0833277i
\(606\) 0 0
\(607\) 9.14071 + 15.8322i 0.371010 + 0.642608i 0.989721 0.143011i \(-0.0456783\pi\)
−0.618711 + 0.785618i \(0.712345\pi\)
\(608\) 0 0
\(609\) 9.12195 11.8296i 0.369640 0.479361i
\(610\) 0 0
\(611\) 2.11158 + 1.43965i 0.0854254 + 0.0582420i
\(612\) 0 0
\(613\) 7.68761 + 7.13306i 0.310500 + 0.288102i 0.819994 0.572373i \(-0.193977\pi\)
−0.509494 + 0.860474i \(0.670167\pi\)
\(614\) 0 0
\(615\) 0.572155 0.275535i 0.0230715 0.0111107i
\(616\) 0 0
\(617\) −30.5676 14.7206i −1.23061 0.592629i −0.298361 0.954453i \(-0.596440\pi\)
−0.932246 + 0.361824i \(0.882154\pi\)
\(618\) 0 0
\(619\) 13.5713 23.5062i 0.545478 0.944795i −0.453099 0.891460i \(-0.649682\pi\)
0.998577 0.0533348i \(-0.0169850\pi\)
\(620\) 0 0
\(621\) −29.3923 4.43017i −1.17947 0.177777i
\(622\) 0 0
\(623\) 0.380953 6.50726i 0.0152625 0.260708i
\(624\) 0 0
\(625\) 1.68795 + 22.5241i 0.0675179 + 0.900964i
\(626\) 0 0
\(627\) −6.62731 16.8861i −0.264669 0.674366i
\(628\) 0 0
\(629\) −25.9532 32.5443i −1.03482 1.29763i
\(630\) 0 0
\(631\) −6.67656 + 8.37214i −0.265790 + 0.333290i −0.896760 0.442517i \(-0.854086\pi\)
0.630970 + 0.775807i \(0.282657\pi\)
\(632\) 0 0
\(633\) −2.53042 + 33.7661i −0.100575 + 1.34208i
\(634\) 0 0
\(635\) 0.362613 0.336456i 0.0143899 0.0133519i
\(636\) 0 0
\(637\) −1.45711 + 2.72556i −0.0577330 + 0.107991i
\(638\) 0 0
\(639\) −6.12267 + 5.68100i −0.242209 + 0.224737i
\(640\) 0 0
\(641\) 1.86964 24.9486i 0.0738463 0.985410i −0.830059 0.557676i \(-0.811693\pi\)
0.903905 0.427734i \(-0.140688\pi\)
\(642\) 0 0
\(643\) 7.67367 9.62248i 0.302620 0.379474i −0.607149 0.794588i \(-0.707687\pi\)
0.909769 + 0.415114i \(0.136258\pi\)
\(644\) 0 0
\(645\) 1.52817 + 1.91626i 0.0601716 + 0.0754528i
\(646\) 0 0
\(647\) 9.77804 + 24.9140i 0.384414 + 0.979472i 0.983841 + 0.179043i \(0.0573000\pi\)
−0.599427 + 0.800430i \(0.704605\pi\)
\(648\) 0 0
\(649\) −0.686944 9.16663i −0.0269649 0.359822i
\(650\) 0 0
\(651\) 25.0153 + 22.4622i 0.980427 + 0.880362i
\(652\) 0 0
\(653\) 27.9398 + 4.21124i 1.09337 + 0.164799i 0.670877 0.741569i \(-0.265918\pi\)
0.422490 + 0.906367i \(0.361156\pi\)
\(654\) 0 0
\(655\) 0.666469 1.15436i 0.0260411 0.0451045i
\(656\) 0 0
\(657\) 5.03050 + 2.42256i 0.196258 + 0.0945130i
\(658\) 0 0
\(659\) 22.1915 10.6869i 0.864459 0.416302i 0.0515351 0.998671i \(-0.483589\pi\)
0.812924 + 0.582370i \(0.197874\pi\)
\(660\) 0 0
\(661\) −27.1359 25.1784i −1.05546 0.979326i −0.0556456 0.998451i \(-0.517722\pi\)
−0.999817 + 0.0191243i \(0.993912\pi\)
\(662\) 0 0
\(663\) 3.64180 + 2.48294i 0.141436 + 0.0964292i
\(664\) 0 0
\(665\) 1.85403 8.78140i 0.0718962 0.340528i
\(666\) 0 0
\(667\) 9.78044 + 16.9402i 0.378700 + 0.655928i
\(668\) 0 0
\(669\) −2.45675 + 1.67499i −0.0949836 + 0.0647587i
\(670\) 0 0
\(671\) −1.32241 + 5.79385i −0.0510510 + 0.223669i
\(672\) 0 0
\(673\) 10.7101 + 46.9240i 0.412844 + 1.80879i 0.570505 + 0.821294i \(0.306748\pi\)
−0.157660 + 0.987493i \(0.550395\pi\)
\(674\) 0 0
\(675\) 25.8616 + 7.97726i 0.995415 + 0.307045i
\(676\) 0 0
\(677\) −18.9609 + 5.84866i −0.728727 + 0.224782i −0.636850 0.770988i \(-0.719763\pi\)
−0.0918770 + 0.995770i \(0.529287\pi\)
\(678\) 0 0
\(679\) −0.642061 + 1.71788i −0.0246400 + 0.0659263i
\(680\) 0 0
\(681\) −6.75050 + 17.2000i −0.258680 + 0.659105i
\(682\) 0 0
\(683\) 7.52385 1.13404i 0.287892 0.0433928i −0.00350850 0.999994i \(-0.501117\pi\)
0.291401 + 0.956601i \(0.405879\pi\)
\(684\) 0 0
\(685\) −4.71926 −0.180314
\(686\) 0 0
\(687\) −22.0424 −0.840968
\(688\) 0 0
\(689\) 4.80402 0.724089i 0.183019 0.0275856i
\(690\) 0 0
\(691\) −7.62823 + 19.4364i −0.290191 + 0.739396i 0.709217 + 0.704991i \(0.249049\pi\)
−0.999408 + 0.0344051i \(0.989046\pi\)
\(692\) 0 0
\(693\) 0.844401 2.25926i 0.0320761 0.0858222i
\(694\) 0 0
\(695\) −4.17950 + 1.28920i −0.158537 + 0.0489023i
\(696\) 0 0
\(697\) 6.38810 + 1.97047i 0.241966 + 0.0746367i
\(698\) 0 0
\(699\) −7.08891 31.0585i −0.268127 1.17474i
\(700\) 0 0
\(701\) 5.08613 22.2838i 0.192100 0.841647i −0.783377 0.621547i \(-0.786504\pi\)
0.975477 0.220100i \(-0.0706384\pi\)
\(702\) 0 0
\(703\) −44.4430 + 30.3007i −1.67620 + 1.14281i
\(704\) 0 0
\(705\) 1.78976 + 3.09995i 0.0674061 + 0.116751i
\(706\) 0 0
\(707\) 3.97653 18.8344i 0.149553 0.708339i
\(708\) 0 0
\(709\) −2.61602 1.78358i −0.0982469 0.0669836i 0.513200 0.858269i \(-0.328460\pi\)
−0.611447 + 0.791285i \(0.709412\pi\)
\(710\) 0 0
\(711\) −4.86207 4.51134i −0.182342 0.169189i
\(712\) 0 0
\(713\) −39.6642 + 19.1013i −1.48544 + 0.715348i
\(714\) 0 0
\(715\) −0.225565 0.108626i −0.00843566 0.00406240i
\(716\) 0 0
\(717\) −15.3368 + 26.5641i −0.572763 + 0.992055i
\(718\) 0 0
\(719\) −0.938530 0.141461i −0.0350012 0.00527559i 0.131518 0.991314i \(-0.458015\pi\)
−0.166520 + 0.986038i \(0.553253\pi\)
\(720\) 0 0
\(721\) −11.6178 10.4321i −0.432671 0.388512i
\(722\) 0 0
\(723\) −0.473564 6.31928i −0.0176120 0.235016i
\(724\) 0 0
\(725\) −6.50680 16.5791i −0.241657 0.615731i
\(726\) 0 0
\(727\) 5.90889 + 7.40951i 0.219148 + 0.274803i 0.879237 0.476384i \(-0.158053\pi\)
−0.660089 + 0.751188i \(0.729481\pi\)
\(728\) 0 0
\(729\) 18.7299 23.4866i 0.693701 0.869873i
\(730\) 0 0
\(731\) −1.92815 + 25.7294i −0.0713153 + 0.951637i
\(732\) 0 0
\(733\) 24.3059 22.5526i 0.897758 0.832998i −0.0888605 0.996044i \(-0.528323\pi\)
0.986618 + 0.163047i \(0.0521321\pi\)
\(734\) 0 0
\(735\) −3.49511 + 2.55394i −0.128919 + 0.0942036i
\(736\) 0 0
\(737\) 8.71473 8.08609i 0.321011 0.297855i
\(738\) 0 0
\(739\) 1.82028 24.2899i 0.0669601 0.893520i −0.858047 0.513571i \(-0.828322\pi\)
0.925007 0.379949i \(-0.124059\pi\)
\(740\) 0 0
\(741\) 3.55120 4.45307i 0.130457 0.163587i
\(742\) 0 0
\(743\) −1.22787 1.53970i −0.0450460 0.0564859i 0.758798 0.651326i \(-0.225787\pi\)
−0.803844 + 0.594840i \(0.797215\pi\)
\(744\) 0 0
\(745\) 2.70254 + 6.88595i 0.0990132 + 0.252282i
\(746\) 0 0
\(747\) −0.305912 4.08211i −0.0111927 0.149357i
\(748\) 0 0
\(749\) −1.64213 + 28.0500i −0.0600019 + 1.02493i
\(750\) 0 0
\(751\) 27.5074 + 4.14607i 1.00376 + 0.151292i 0.630298 0.776354i \(-0.282933\pi\)
0.373461 + 0.927646i \(0.378171\pi\)
\(752\) 0 0
\(753\) 14.8954 25.7996i 0.542818 0.940189i
\(754\) 0 0
\(755\) 4.21322 + 2.02898i 0.153335 + 0.0738421i
\(756\) 0 0
\(757\) 11.7753 5.67068i 0.427980 0.206104i −0.207478 0.978240i \(-0.566526\pi\)
0.635458 + 0.772135i \(0.280811\pi\)
\(758\) 0 0
\(759\) −8.39838 7.79256i −0.304842 0.282852i
\(760\) 0 0
\(761\) −1.21595 0.829020i −0.0440781 0.0300520i 0.541081 0.840971i \(-0.318015\pi\)
−0.585159 + 0.810919i \(0.698968\pi\)
\(762\) 0 0
\(763\) 8.52937 11.0612i 0.308784 0.400442i
\(764\) 0 0
\(765\) −0.850923 1.47384i −0.0307652 0.0532869i
\(766\) 0 0
\(767\) 2.38473 1.62588i 0.0861075 0.0587071i
\(768\) 0 0
\(769\) 1.93322 8.46998i 0.0697136 0.305435i −0.928036 0.372491i \(-0.878504\pi\)
0.997749 + 0.0670563i \(0.0213607\pi\)
\(770\) 0 0
\(771\) 8.82084 + 38.6466i 0.317675 + 1.39182i
\(772\) 0 0
\(773\) 22.8028 + 7.03374i 0.820161 + 0.252986i 0.676314 0.736614i \(-0.263576\pi\)
0.143847 + 0.989600i \(0.454053\pi\)
\(774\) 0 0
\(775\) 38.3031 11.8149i 1.37589 0.424405i
\(776\) 0 0
\(777\) 25.7135 + 3.44743i 0.922468 + 0.123676i
\(778\) 0 0
\(779\) 3.15604 8.04145i 0.113077 0.288115i
\(780\) 0 0
\(781\) −17.9141 + 2.70012i −0.641018 + 0.0966179i
\(782\) 0 0
\(783\) −20.5987 −0.736137
\(784\) 0 0
\(785\) 8.25592 0.294666
\(786\) 0 0
\(787\) −38.5373 + 5.80856i −1.37371 + 0.207053i −0.794072 0.607824i \(-0.792043\pi\)
−0.579634 + 0.814877i \(0.696805\pi\)
\(788\) 0 0
\(789\) −11.3042 + 28.8028i −0.402442 + 1.02541i
\(790\) 0 0
\(791\) −12.9352 8.51313i −0.459922 0.302692i
\(792\) 0 0
\(793\) −1.78306 + 0.550001i −0.0633183 + 0.0195311i
\(794\) 0 0
\(795\) 6.50232 + 2.00570i 0.230614 + 0.0711349i
\(796\) 0 0
\(797\) 9.21253 + 40.3627i 0.326325 + 1.42972i 0.826079 + 0.563554i \(0.190566\pi\)
−0.499755 + 0.866167i \(0.666576\pi\)
\(798\) 0 0
\(799\) −8.38492 + 36.7367i −0.296637 + 1.29965i
\(800\) 0 0
\(801\) −1.31968 + 0.899743i −0.0466286 + 0.0317909i
\(802\) 0 0
\(803\) 6.05535 + 10.4882i 0.213689 + 0.370119i
\(804\) 0 0
\(805\) −1.35136 5.50490i −0.0476292 0.194022i
\(806\) 0 0
\(807\) 34.6093 + 23.5963i 1.21831 + 0.830627i
\(808\) 0 0
\(809\) −10.0710 9.34455i −0.354079 0.328537i 0.482999 0.875621i \(-0.339547\pi\)
−0.837078 + 0.547084i \(0.815738\pi\)
\(810\) 0 0
\(811\) 18.7641 9.03630i 0.658896 0.317307i −0.0743888 0.997229i \(-0.523701\pi\)
0.733285 + 0.679922i \(0.237986\pi\)
\(812\) 0 0
\(813\) −42.8003 20.6115i −1.50107 0.722878i
\(814\) 0 0
\(815\) 0.514721 0.891523i 0.0180299 0.0312287i
\(816\) 0 0
\(817\) 32.9688 + 4.96925i 1.15343 + 0.173852i
\(818\) 0 0
\(819\) 0.746897 0.125078i 0.0260987 0.00437057i
\(820\) 0 0
\(821\) 3.54655 + 47.3254i 0.123775 + 1.65167i 0.620828 + 0.783947i \(0.286797\pi\)
−0.497053 + 0.867720i \(0.665584\pi\)
\(822\) 0 0
\(823\) 9.84217 + 25.0775i 0.343077 + 0.874145i 0.993422 + 0.114507i \(0.0365287\pi\)
−0.650346 + 0.759638i \(0.725376\pi\)
\(824\) 0 0
\(825\) 6.50386 + 8.15558i 0.226435 + 0.283941i
\(826\) 0 0
\(827\) 28.8554 36.1835i 1.00340 1.25822i 0.0375040 0.999296i \(-0.488059\pi\)
0.965897 0.258928i \(-0.0833693\pi\)
\(828\) 0 0
\(829\) −0.0884937 + 1.18087i −0.00307351 + 0.0410132i −0.998530 0.0542077i \(-0.982737\pi\)
0.995456 + 0.0952209i \(0.0303557\pi\)
\(830\) 0 0
\(831\) −15.4058 + 14.2945i −0.534420 + 0.495869i
\(832\) 0 0
\(833\) −45.2578 5.31725i −1.56809 0.184232i
\(834\) 0 0
\(835\) −3.70125 + 3.43426i −0.128087 + 0.118848i
\(836\) 0 0
\(837\) 3.46446 46.2300i 0.119749 1.59794i
\(838\) 0 0
\(839\) −10.2114 + 12.8047i −0.352537 + 0.442068i −0.926205 0.377020i \(-0.876949\pi\)
0.573668 + 0.819088i \(0.305520\pi\)
\(840\) 0 0
\(841\) −9.62947 12.0750i −0.332051 0.416378i
\(842\) 0 0
\(843\) −17.3813 44.2868i −0.598644 1.52532i
\(844\) 0 0
\(845\) 0.385882 + 5.14923i 0.0132747 + 0.177139i
\(846\) 0 0
\(847\) −19.5016 + 13.7676i −0.670084 + 0.473060i
\(848\) 0 0
\(849\) −47.9768 7.23134i −1.64656 0.248179i
\(850\) 0 0
\(851\) −16.9859 + 29.4205i −0.582271 + 1.00852i
\(852\) 0 0
\(853\) 39.6712 + 19.1046i 1.35832 + 0.654131i 0.964261 0.264956i \(-0.0853574\pi\)
0.394056 + 0.919086i \(0.371072\pi\)
\(854\) 0 0
\(855\) −1.98137 + 0.954178i −0.0677615 + 0.0326322i
\(856\) 0 0
\(857\) −5.55267 5.15212i −0.189675 0.175993i 0.579619 0.814888i \(-0.303201\pi\)
−0.769294 + 0.638895i \(0.779392\pi\)
\(858\) 0 0
\(859\) 16.3222 + 11.1283i 0.556907 + 0.379693i 0.808819 0.588057i \(-0.200107\pi\)
−0.251912 + 0.967750i \(0.581059\pi\)
\(860\) 0 0
\(861\) −3.72392 + 1.86883i −0.126911 + 0.0636896i
\(862\) 0 0
\(863\) −11.7008 20.2663i −0.398299 0.689874i 0.595217 0.803565i \(-0.297066\pi\)
−0.993516 + 0.113691i \(0.963733\pi\)
\(864\) 0 0
\(865\) −2.68499 + 1.83059i −0.0912923 + 0.0622420i
\(866\) 0 0
\(867\) −8.66018 + 37.9427i −0.294115 + 1.28860i
\(868\) 0 0
\(869\) −3.20129 14.0258i −0.108596 0.475792i
\(870\) 0 0
\(871\) 3.56689 + 1.10024i 0.120860 + 0.0372802i
\(872\) 0 0
\(873\) 0.429411 0.132456i 0.0145334 0.00448295i
\(874\) 0 0
\(875\) 0.955071 + 10.4520i 0.0322873 + 0.353343i
\(876\) 0 0
\(877\) 3.75483 9.56715i 0.126792 0.323060i −0.853356 0.521329i \(-0.825437\pi\)
0.980148 + 0.198269i \(0.0635319\pi\)
\(878\) 0 0
\(879\) −7.15276 + 1.07811i −0.241257 + 0.0363636i
\(880\) 0 0
\(881\) −35.8408 −1.20751 −0.603754 0.797171i \(-0.706329\pi\)
−0.603754 + 0.797171i \(0.706329\pi\)
\(882\) 0 0
\(883\) 17.3715 0.584596 0.292298 0.956327i \(-0.405580\pi\)
0.292298 + 0.956327i \(0.405580\pi\)
\(884\) 0 0
\(885\) 3.99740 0.602510i 0.134371 0.0202532i
\(886\) 0 0
\(887\) 10.0573 25.6257i 0.337693 0.860427i −0.656649 0.754196i \(-0.728027\pi\)
0.994342 0.106230i \(-0.0338781\pi\)
\(888\) 0 0
\(889\) −2.34277 + 2.24604i −0.0785739 + 0.0753298i
\(890\) 0 0
\(891\) 8.91525 2.74999i 0.298672 0.0921281i
\(892\) 0 0
\(893\) 46.5294 + 14.3524i 1.55705 + 0.480285i
\(894\) 0 0
\(895\) −0.449043 1.96739i −0.0150098 0.0657624i
\(896\) 0 0
\(897\) 0.800458 3.50704i 0.0267265 0.117097i
\(898\) 0 0
\(899\) −25.2071 + 17.1859i −0.840703 + 0.573182i
\(900\) 0 0
\(901\) 35.8161 + 62.0354i 1.19321 + 2.06670i
\(902\) 0 0
\(903\) −10.2303 12.4073i −0.340442 0.412890i
\(904\) 0 0
\(905\) 4.86013 + 3.31358i 0.161556 + 0.110147i
\(906\) 0 0
\(907\) 2.34689 + 2.17759i 0.0779270 + 0.0723057i 0.718175 0.695863i \(-0.244978\pi\)
−0.640248 + 0.768169i \(0.721168\pi\)
\(908\) 0 0
\(909\) −4.24965 + 2.04652i −0.140952 + 0.0678789i
\(910\) 0 0
\(911\) 28.9643 + 13.9485i 0.959631 + 0.462134i 0.847053 0.531509i \(-0.178375\pi\)
0.112579 + 0.993643i \(0.464089\pi\)
\(912\) 0 0
\(913\) 4.43955 7.68953i 0.146928 0.254486i
\(914\) 0 0
\(915\) −2.58432 0.389523i −0.0854349 0.0128772i
\(916\) 0 0
\(917\) −4.24852 + 7.64415i −0.140298 + 0.252432i
\(918\) 0 0
\(919\) 2.16230 + 28.8539i 0.0713278 + 0.951803i 0.911982 + 0.410231i \(0.134552\pi\)
−0.840654 + 0.541573i \(0.817829\pi\)
\(920\) 0 0
\(921\) −4.01493 10.2299i −0.132297 0.337086i
\(922\) 0 0
\(923\) −3.54658 4.44727i −0.116737 0.146384i
\(924\) 0 0
\(925\) 19.2855 24.1832i 0.634102 0.795139i
\(926\) 0 0
\(927\) −0.285915 + 3.81528i −0.00939069 + 0.125310i
\(928\) 0 0
\(929\) −10.9198 + 10.1321i −0.358267 + 0.332423i −0.838685 0.544616i \(-0.816675\pi\)
0.480418 + 0.877040i \(0.340485\pi\)
\(930\) 0 0
\(931\) −11.2223 + 57.8059i −0.367795 + 1.89451i
\(932\) 0 0
\(933\) 1.93547 1.79585i 0.0633645 0.0587936i
\(934\) 0 0
\(935\) 0.275857 3.68105i 0.00902149 0.120383i
\(936\) 0 0
\(937\) −31.8518 + 39.9409i −1.04055 + 1.30481i −0.0894329 + 0.995993i \(0.528505\pi\)
−0.951121 + 0.308820i \(0.900066\pi\)
\(938\) 0 0
\(939\) −9.22091 11.5627i −0.300913 0.377333i
\(940\) 0 0
\(941\) −5.60138 14.2721i −0.182600 0.465257i 0.810038 0.586377i \(-0.199446\pi\)
−0.992638 + 0.121121i \(0.961351\pi\)
\(942\) 0 0
\(943\) −0.407719 5.44063i −0.0132772 0.177171i
\(944\) 0 0
\(945\) 5.73185 + 1.66608i 0.186457 + 0.0541975i
\(946\) 0 0
\(947\) 4.57968 + 0.690276i 0.148820 + 0.0224310i 0.223030 0.974812i \(-0.428405\pi\)
−0.0742098 + 0.997243i \(0.523643\pi\)
\(948\) 0 0
\(949\) −1.90128 + 3.29311i −0.0617181 + 0.106899i
\(950\) 0 0
\(951\) −11.1757 5.38192i −0.362396 0.174521i
\(952\) 0 0
\(953\) −53.0768 + 25.5604i −1.71933 + 0.827984i −0.729794 + 0.683667i \(0.760384\pi\)
−0.989532 + 0.144317i \(0.953902\pi\)
\(954\) 0 0
\(955\) −6.33898 5.88171i −0.205125 0.190328i
\(956\) 0 0
\(957\) −6.55986 4.47244i −0.212050 0.144573i
\(958\) 0 0
\(959\) 30.9591 0.505416i 0.999722 0.0163207i
\(960\) 0 0
\(961\) −18.8311 32.6164i −0.607455 1.05214i
\(962\) 0 0
\(963\) 5.68858 3.87841i 0.183312 0.124980i
\(964\) 0 0
\(965\) 1.83432 8.03666i 0.0590487 0.258709i
\(966\) 0 0
\(967\) 13.2671 + 58.1271i 0.426642 + 1.86924i 0.490755 + 0.871297i \(0.336721\pi\)
−0.0641134 + 0.997943i \(0.520422\pi\)
\(968\) 0 0
\(969\) 80.2483 + 24.7533i 2.57795 + 0.795191i
\(970\) 0 0
\(971\) 12.7435 3.93084i 0.408958 0.126147i −0.0834487 0.996512i \(-0.526593\pi\)
0.492407 + 0.870365i \(0.336117\pi\)
\(972\) 0 0
\(973\) 27.2801 8.90499i 0.874559 0.285481i
\(974\) 0 0
\(975\) −1.19659 + 3.04887i −0.0383217 + 0.0976421i
\(976\) 0 0
\(977\) −24.3607 + 3.67179i −0.779369 + 0.117471i −0.526665 0.850073i \(-0.676558\pi\)
−0.252704 + 0.967544i \(0.581320\pi\)
\(978\) 0 0
\(979\) −3.46443 −0.110724
\(980\) 0 0
\(981\) −3.42256 −0.109274
\(982\) 0 0
\(983\) −22.3623 + 3.37057i −0.713245 + 0.107504i −0.495634 0.868531i \(-0.665064\pi\)
−0.217611 + 0.976036i \(0.569826\pi\)
\(984\) 0 0
\(985\) −0.757915 + 1.93114i −0.0241492 + 0.0615311i
\(986\) 0 0
\(987\) −12.0731 20.1445i −0.384291 0.641206i
\(988\) 0 0
\(989\) 20.1218 6.20676i 0.639837 0.197364i
\(990\) 0 0
\(991\) −52.2037 16.1027i −1.65830 0.511519i −0.682633 0.730761i \(-0.739165\pi\)
−0.975670 + 0.219242i \(0.929642\pi\)
\(992\) 0 0
\(993\) 9.40885 + 41.2229i 0.298581 + 1.30817i
\(994\) 0 0
\(995\) 0.468937 2.05455i 0.0148663 0.0651336i
\(996\) 0 0
\(997\) −17.2173 + 11.7386i −0.545277 + 0.371764i −0.804407 0.594079i \(-0.797517\pi\)
0.259130 + 0.965843i \(0.416564\pi\)
\(998\) 0 0
\(999\) −17.8871 30.9814i −0.565924 0.980209i
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 392.2.y.a.305.3 yes 84
4.3 odd 2 784.2.bg.f.305.5 84
49.9 even 21 inner 392.2.y.a.9.3 84
196.107 odd 42 784.2.bg.f.401.5 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
392.2.y.a.9.3 84 49.9 even 21 inner
392.2.y.a.305.3 yes 84 1.1 even 1 trivial
784.2.bg.f.305.5 84 4.3 odd 2
784.2.bg.f.401.5 84 196.107 odd 42