Properties

Label 864.2.y.c.97.9
Level $864$
Weight $2$
Character 864.97
Analytic conductor $6.899$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [864,2,Mod(97,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.y (of order \(9\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 97.9
Character \(\chi\) \(=\) 864.97
Dual form 864.2.y.c.481.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.68626 + 0.395655i) q^{3} +(-2.65901 - 0.967801i) q^{5} +(0.872194 - 4.94646i) q^{7} +(2.68691 + 1.33435i) q^{9} +O(q^{10})\) \(q+(1.68626 + 0.395655i) q^{3} +(-2.65901 - 0.967801i) q^{5} +(0.872194 - 4.94646i) q^{7} +(2.68691 + 1.33435i) q^{9} +(-4.78983 + 1.74335i) q^{11} +(-1.62758 + 1.36570i) q^{13} +(-4.10086 - 2.68401i) q^{15} +(-2.73689 - 4.74043i) q^{17} +(-0.0248312 + 0.0430090i) q^{19} +(3.42783 - 7.99590i) q^{21} +(-0.922436 - 5.23139i) q^{23} +(2.30348 + 1.93285i) q^{25} +(4.00288 + 3.31315i) q^{27} +(-4.38054 - 3.67571i) q^{29} +(1.15756 + 6.56488i) q^{31} +(-8.76664 + 1.04462i) q^{33} +(-7.10636 + 12.3086i) q^{35} +(-1.73838 - 3.01096i) q^{37} +(-3.28487 + 1.65896i) q^{39} +(-0.431714 + 0.362251i) q^{41} +(8.90987 - 3.24293i) q^{43} +(-5.85315 - 6.14845i) q^{45} +(-0.332415 + 1.88522i) q^{47} +(-17.1289 - 6.23440i) q^{49} +(-2.73952 - 9.07644i) q^{51} +6.01999 q^{53} +14.4234 q^{55} +(-0.0588885 + 0.0626995i) q^{57} +(0.00945632 + 0.00344182i) q^{59} +(1.68535 - 9.55809i) q^{61} +(8.94382 - 12.1269i) q^{63} +(5.64949 - 2.05625i) q^{65} +(-2.39581 + 2.01032i) q^{67} +(0.514367 - 9.18643i) q^{69} +(3.24130 + 5.61409i) q^{71} +(3.15649 - 5.46720i) q^{73} +(3.11951 + 4.17066i) q^{75} +(4.44577 + 25.2132i) q^{77} +(-4.46970 - 3.75052i) q^{79} +(5.43901 + 7.17058i) q^{81} +(3.18532 + 2.67280i) q^{83} +(2.68962 + 15.2536i) q^{85} +(-5.93240 - 7.93137i) q^{87} +(4.80540 - 8.32321i) q^{89} +(5.33583 + 9.24193i) q^{91} +(-0.645479 + 11.5281i) q^{93} +(0.107651 - 0.0903296i) q^{95} +(15.9754 - 5.81458i) q^{97} +(-15.1961 - 1.70707i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 54 q+O(q^{10}) \) Copy content Toggle raw display \( 54 q + 9 q^{11} + 12 q^{17} - 18 q^{19} + 12 q^{21} + 21 q^{27} + 6 q^{29} - 36 q^{31} - 9 q^{33} - 24 q^{39} + 3 q^{41} + 21 q^{43} + 42 q^{45} - 18 q^{49} - 24 q^{51} + 36 q^{53} + 72 q^{55} + 39 q^{57} - 18 q^{59} - 18 q^{61} + 30 q^{63} + 48 q^{65} + 27 q^{67} + 24 q^{69} + 84 q^{75} + 36 q^{77} - 72 q^{79} + 36 q^{81} - 6 q^{87} + 33 q^{89} - 36 q^{91} + 72 q^{93} - 36 q^{95} + 9 q^{97} - 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{9}\right)\) \(1\)

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
\(3\) 1.68626 + 0.395655i 0.973560 + 0.228432i
\(4\) 0 0
\(5\) −2.65901 0.967801i −1.18915 0.432814i −0.329722 0.944078i \(-0.606955\pi\)
−0.859424 + 0.511264i \(0.829177\pi\)
\(6\) 0 0
\(7\) 0.872194 4.94646i 0.329658 1.86959i −0.145024 0.989428i \(-0.546326\pi\)
0.474683 0.880157i \(-0.342563\pi\)
\(8\) 0 0
\(9\) 2.68691 + 1.33435i 0.895638 + 0.444784i
\(10\) 0 0
\(11\) −4.78983 + 1.74335i −1.44419 + 0.525641i −0.940961 0.338514i \(-0.890076\pi\)
−0.503226 + 0.864155i \(0.667854\pi\)
\(12\) 0 0
\(13\) −1.62758 + 1.36570i −0.451410 + 0.378778i −0.839959 0.542650i \(-0.817421\pi\)
0.388549 + 0.921428i \(0.372977\pi\)
\(14\) 0 0
\(15\) −4.10086 2.68401i −1.05884 0.693009i
\(16\) 0 0
\(17\) −2.73689 4.74043i −0.663793 1.14972i −0.979611 0.200903i \(-0.935612\pi\)
0.315818 0.948820i \(-0.397721\pi\)
\(18\) 0 0
\(19\) −0.0248312 + 0.0430090i −0.00569667 + 0.00986693i −0.868860 0.495058i \(-0.835147\pi\)
0.863163 + 0.504925i \(0.168480\pi\)
\(20\) 0 0
\(21\) 3.42783 7.99590i 0.748015 1.74485i
\(22\) 0 0
\(23\) −0.922436 5.23139i −0.192341 1.09082i −0.916155 0.400825i \(-0.868723\pi\)
0.723813 0.689996i \(-0.242388\pi\)
\(24\) 0 0
\(25\) 2.30348 + 1.93285i 0.460696 + 0.386570i
\(26\) 0 0
\(27\) 4.00288 + 3.31315i 0.770354 + 0.637616i
\(28\) 0 0
\(29\) −4.38054 3.67571i −0.813446 0.682562i 0.137982 0.990435i \(-0.455939\pi\)
−0.951428 + 0.307873i \(0.900383\pi\)
\(30\) 0 0
\(31\) 1.15756 + 6.56488i 0.207905 + 1.17909i 0.892803 + 0.450448i \(0.148736\pi\)
−0.684898 + 0.728639i \(0.740153\pi\)
\(32\) 0 0
\(33\) −8.76664 + 1.04462i −1.52608 + 0.181845i
\(34\) 0 0
\(35\) −7.10636 + 12.3086i −1.20119 + 2.08053i
\(36\) 0 0
\(37\) −1.73838 3.01096i −0.285788 0.494999i 0.687012 0.726646i \(-0.258922\pi\)
−0.972800 + 0.231647i \(0.925589\pi\)
\(38\) 0 0
\(39\) −3.28487 + 1.65896i −0.526000 + 0.265647i
\(40\) 0 0
\(41\) −0.431714 + 0.362251i −0.0674224 + 0.0565741i −0.675876 0.737015i \(-0.736235\pi\)
0.608454 + 0.793589i \(0.291790\pi\)
\(42\) 0 0
\(43\) 8.90987 3.24293i 1.35874 0.494542i 0.443077 0.896484i \(-0.353887\pi\)
0.915665 + 0.401942i \(0.131665\pi\)
\(44\) 0 0
\(45\) −5.85315 6.14845i −0.872536 0.916557i
\(46\) 0 0
\(47\) −0.332415 + 1.88522i −0.0484876 + 0.274987i −0.999406 0.0344539i \(-0.989031\pi\)
0.950919 + 0.309441i \(0.100142\pi\)
\(48\) 0 0
\(49\) −17.1289 6.23440i −2.44698 0.890628i
\(50\) 0 0
\(51\) −2.73952 9.07644i −0.383609 1.27096i
\(52\) 0 0
\(53\) 6.01999 0.826910 0.413455 0.910525i \(-0.364322\pi\)
0.413455 + 0.910525i \(0.364322\pi\)
\(54\) 0 0
\(55\) 14.4234 1.94485
\(56\) 0 0
\(57\) −0.0588885 + 0.0626995i −0.00779997 + 0.00830475i
\(58\) 0 0
\(59\) 0.00945632 + 0.00344182i 0.00123111 + 0.000448087i 0.342636 0.939468i \(-0.388680\pi\)
−0.341405 + 0.939916i \(0.610903\pi\)
\(60\) 0 0
\(61\) 1.68535 9.55809i 0.215787 1.22379i −0.663749 0.747956i \(-0.731036\pi\)
0.879536 0.475833i \(-0.157853\pi\)
\(62\) 0 0
\(63\) 8.94382 12.1269i 1.12682 1.52784i
\(64\) 0 0
\(65\) 5.64949 2.05625i 0.700733 0.255046i
\(66\) 0 0
\(67\) −2.39581 + 2.01032i −0.292694 + 0.245600i −0.777296 0.629135i \(-0.783409\pi\)
0.484601 + 0.874735i \(0.338965\pi\)
\(68\) 0 0
\(69\) 0.514367 9.18643i 0.0619225 1.10592i
\(70\) 0 0
\(71\) 3.24130 + 5.61409i 0.384671 + 0.666270i 0.991724 0.128392i \(-0.0409815\pi\)
−0.607052 + 0.794662i \(0.707648\pi\)
\(72\) 0 0
\(73\) 3.15649 5.46720i 0.369439 0.639887i −0.620039 0.784571i \(-0.712883\pi\)
0.989478 + 0.144684i \(0.0462165\pi\)
\(74\) 0 0
\(75\) 3.11951 + 4.17066i 0.360210 + 0.481586i
\(76\) 0 0
\(77\) 4.44577 + 25.2132i 0.506643 + 2.87331i
\(78\) 0 0
\(79\) −4.46970 3.75052i −0.502881 0.421967i 0.355735 0.934587i \(-0.384231\pi\)
−0.858616 + 0.512620i \(0.828675\pi\)
\(80\) 0 0
\(81\) 5.43901 + 7.17058i 0.604334 + 0.796731i
\(82\) 0 0
\(83\) 3.18532 + 2.67280i 0.349634 + 0.293378i 0.800643 0.599142i \(-0.204491\pi\)
−0.451009 + 0.892519i \(0.648936\pi\)
\(84\) 0 0
\(85\) 2.68962 + 15.2536i 0.291731 + 1.65449i
\(86\) 0 0
\(87\) −5.93240 7.93137i −0.636020 0.850332i
\(88\) 0 0
\(89\) 4.80540 8.32321i 0.509372 0.882258i −0.490569 0.871402i \(-0.663211\pi\)
0.999941 0.0108557i \(-0.00345556\pi\)
\(90\) 0 0
\(91\) 5.33583 + 9.24193i 0.559347 + 0.968817i
\(92\) 0 0
\(93\) −0.645479 + 11.5281i −0.0669330 + 1.19540i
\(94\) 0 0
\(95\) 0.107651 0.0903296i 0.0110447 0.00926762i
\(96\) 0 0
\(97\) 15.9754 5.81458i 1.62206 0.590381i 0.638286 0.769799i \(-0.279644\pi\)
0.983773 + 0.179418i \(0.0574216\pi\)
\(98\) 0 0
\(99\) −15.1961 1.70707i −1.52727 0.171567i
\(100\) 0 0
\(101\) 1.87917 10.6573i 0.186984 1.06044i −0.736395 0.676552i \(-0.763473\pi\)
0.923379 0.383889i \(-0.125415\pi\)
\(102\) 0 0
\(103\) 6.76943 + 2.46387i 0.667012 + 0.242772i 0.653261 0.757133i \(-0.273401\pi\)
0.0137509 + 0.999905i \(0.495623\pi\)
\(104\) 0 0
\(105\) −16.8531 + 17.9437i −1.64469 + 1.75113i
\(106\) 0 0
\(107\) 7.49975 0.725028 0.362514 0.931978i \(-0.381919\pi\)
0.362514 + 0.931978i \(0.381919\pi\)
\(108\) 0 0
\(109\) −5.24898 −0.502761 −0.251380 0.967888i \(-0.580885\pi\)
−0.251380 + 0.967888i \(0.580885\pi\)
\(110\) 0 0
\(111\) −1.74005 5.76505i −0.165158 0.547195i
\(112\) 0 0
\(113\) −7.98635 2.90679i −0.751293 0.273448i −0.0621431 0.998067i \(-0.519794\pi\)
−0.689150 + 0.724619i \(0.742016\pi\)
\(114\) 0 0
\(115\) −2.61018 + 14.8031i −0.243401 + 1.38039i
\(116\) 0 0
\(117\) −6.19551 + 1.49776i −0.572775 + 0.138468i
\(118\) 0 0
\(119\) −25.8354 + 9.40333i −2.36833 + 0.862001i
\(120\) 0 0
\(121\) 11.4767 9.63007i 1.04333 0.875461i
\(122\) 0 0
\(123\) −0.871306 + 0.440037i −0.0785630 + 0.0396768i
\(124\) 0 0
\(125\) 2.81979 + 4.88401i 0.252209 + 0.436839i
\(126\) 0 0
\(127\) −3.20983 + 5.55960i −0.284827 + 0.493334i −0.972567 0.232622i \(-0.925269\pi\)
0.687740 + 0.725957i \(0.258603\pi\)
\(128\) 0 0
\(129\) 16.3074 1.94317i 1.43579 0.171086i
\(130\) 0 0
\(131\) 3.34360 + 18.9625i 0.292132 + 1.65676i 0.678636 + 0.734475i \(0.262571\pi\)
−0.386504 + 0.922288i \(0.626317\pi\)
\(132\) 0 0
\(133\) 0.191084 + 0.160339i 0.0165691 + 0.0139031i
\(134\) 0 0
\(135\) −7.43723 12.6837i −0.640095 1.09164i
\(136\) 0 0
\(137\) 1.76113 + 1.47777i 0.150464 + 0.126254i 0.714912 0.699214i \(-0.246467\pi\)
−0.564448 + 0.825468i \(0.690911\pi\)
\(138\) 0 0
\(139\) −1.46944 8.33360i −0.124636 0.706847i −0.981523 0.191343i \(-0.938716\pi\)
0.856887 0.515504i \(-0.172395\pi\)
\(140\) 0 0
\(141\) −1.30643 + 3.04743i −0.110021 + 0.256640i
\(142\) 0 0
\(143\) 5.41493 9.37894i 0.452820 0.784307i
\(144\) 0 0
\(145\) 8.09055 + 14.0132i 0.671884 + 1.16374i
\(146\) 0 0
\(147\) −26.4170 17.2899i −2.17884 1.42605i
\(148\) 0 0
\(149\) −5.13012 + 4.30468i −0.420276 + 0.352653i −0.828268 0.560332i \(-0.810673\pi\)
0.407992 + 0.912985i \(0.366229\pi\)
\(150\) 0 0
\(151\) 2.43651 0.886818i 0.198281 0.0721682i −0.240971 0.970532i \(-0.577466\pi\)
0.439252 + 0.898364i \(0.355244\pi\)
\(152\) 0 0
\(153\) −1.02838 16.3891i −0.0831397 1.32498i
\(154\) 0 0
\(155\) 3.27552 18.5764i 0.263096 1.49209i
\(156\) 0 0
\(157\) −20.5923 7.49499i −1.64344 0.598165i −0.655808 0.754927i \(-0.727672\pi\)
−0.987636 + 0.156762i \(0.949894\pi\)
\(158\) 0 0
\(159\) 10.1512 + 2.38184i 0.805046 + 0.188892i
\(160\) 0 0
\(161\) −26.6814 −2.10279
\(162\) 0 0
\(163\) 7.50928 0.588173 0.294086 0.955779i \(-0.404985\pi\)
0.294086 + 0.955779i \(0.404985\pi\)
\(164\) 0 0
\(165\) 24.3216 + 5.70671i 1.89343 + 0.444266i
\(166\) 0 0
\(167\) −2.31112 0.841178i −0.178840 0.0650923i 0.251048 0.967975i \(-0.419225\pi\)
−0.429888 + 0.902882i \(0.641447\pi\)
\(168\) 0 0
\(169\) −1.47355 + 8.35691i −0.113350 + 0.642839i
\(170\) 0 0
\(171\) −0.124108 + 0.0824277i −0.00949081 + 0.00630341i
\(172\) 0 0
\(173\) 7.62695 2.77598i 0.579866 0.211054i −0.0354003 0.999373i \(-0.511271\pi\)
0.615267 + 0.788319i \(0.289048\pi\)
\(174\) 0 0
\(175\) 11.5698 9.70825i 0.874597 0.733874i
\(176\) 0 0
\(177\) 0.0145840 + 0.00954523i 0.00109620 + 0.000717463i
\(178\) 0 0
\(179\) −6.95175 12.0408i −0.519599 0.899971i −0.999740 0.0227802i \(-0.992748\pi\)
0.480142 0.877191i \(-0.340585\pi\)
\(180\) 0 0
\(181\) −11.9558 + 20.7081i −0.888670 + 1.53922i −0.0472210 + 0.998884i \(0.515037\pi\)
−0.841449 + 0.540337i \(0.818297\pi\)
\(182\) 0 0
\(183\) 6.62364 15.4506i 0.489634 1.14214i
\(184\) 0 0
\(185\) 1.70836 + 9.68859i 0.125601 + 0.712319i
\(186\) 0 0
\(187\) 21.3735 + 17.9345i 1.56298 + 1.31150i
\(188\) 0 0
\(189\) 19.8796 16.9104i 1.44603 1.23005i
\(190\) 0 0
\(191\) 16.9180 + 14.1959i 1.22414 + 1.02718i 0.998597 + 0.0529455i \(0.0168610\pi\)
0.225546 + 0.974233i \(0.427583\pi\)
\(192\) 0 0
\(193\) 4.31347 + 24.4629i 0.310490 + 1.76088i 0.596463 + 0.802640i \(0.296572\pi\)
−0.285973 + 0.958238i \(0.592317\pi\)
\(194\) 0 0
\(195\) 10.3400 1.23211i 0.740466 0.0882329i
\(196\) 0 0
\(197\) 1.84142 3.18944i 0.131196 0.227238i −0.792942 0.609297i \(-0.791452\pi\)
0.924138 + 0.382059i \(0.124785\pi\)
\(198\) 0 0
\(199\) −12.7419 22.0696i −0.903248 1.56447i −0.823252 0.567676i \(-0.807843\pi\)
−0.0799957 0.996795i \(-0.525491\pi\)
\(200\) 0 0
\(201\) −4.83533 + 2.44200i −0.341058 + 0.172245i
\(202\) 0 0
\(203\) −22.0024 + 18.4622i −1.54427 + 1.29579i
\(204\) 0 0
\(205\) 1.49852 0.545416i 0.104661 0.0380935i
\(206\) 0 0
\(207\) 4.50201 15.2872i 0.312912 1.06253i
\(208\) 0 0
\(209\) 0.0439575 0.249295i 0.00304060 0.0172441i
\(210\) 0 0
\(211\) 1.96210 + 0.714147i 0.135077 + 0.0491639i 0.408674 0.912680i \(-0.365991\pi\)
−0.273597 + 0.961844i \(0.588214\pi\)
\(212\) 0 0
\(213\) 3.24441 + 10.7492i 0.222303 + 0.736525i
\(214\) 0 0
\(215\) −26.8299 −1.82979
\(216\) 0 0
\(217\) 33.4825 2.27294
\(218\) 0 0
\(219\) 7.48577 7.97021i 0.505842 0.538577i
\(220\) 0 0
\(221\) 10.9285 + 3.97766i 0.735133 + 0.267566i
\(222\) 0 0
\(223\) −0.149084 + 0.845499i −0.00998342 + 0.0566188i −0.989392 0.145267i \(-0.953596\pi\)
0.979409 + 0.201886i \(0.0647070\pi\)
\(224\) 0 0
\(225\) 3.61015 + 8.26705i 0.240677 + 0.551137i
\(226\) 0 0
\(227\) 6.80525 2.47691i 0.451680 0.164398i −0.106156 0.994350i \(-0.533854\pi\)
0.557836 + 0.829951i \(0.311632\pi\)
\(228\) 0 0
\(229\) −12.2753 + 10.3002i −0.811173 + 0.680655i −0.950887 0.309537i \(-0.899826\pi\)
0.139714 + 0.990192i \(0.455382\pi\)
\(230\) 0 0
\(231\) −2.47904 + 44.2749i −0.163109 + 2.91308i
\(232\) 0 0
\(233\) −11.8424 20.5116i −0.775820 1.34376i −0.934333 0.356402i \(-0.884003\pi\)
0.158513 0.987357i \(-0.449330\pi\)
\(234\) 0 0
\(235\) 2.70841 4.69110i 0.176677 0.306014i
\(236\) 0 0
\(237\) −6.05314 8.09280i −0.393194 0.525684i
\(238\) 0 0
\(239\) 2.34274 + 13.2864i 0.151539 + 0.859423i 0.961882 + 0.273466i \(0.0881701\pi\)
−0.810342 + 0.585957i \(0.800719\pi\)
\(240\) 0 0
\(241\) −1.72399 1.44660i −0.111052 0.0931838i 0.585571 0.810621i \(-0.300870\pi\)
−0.696623 + 0.717438i \(0.745315\pi\)
\(242\) 0 0
\(243\) 6.33448 + 14.2434i 0.406357 + 0.913714i
\(244\) 0 0
\(245\) 39.5122 + 33.1547i 2.52434 + 2.11817i
\(246\) 0 0
\(247\) −0.0183226 0.103913i −0.00116584 0.00661181i
\(248\) 0 0
\(249\) 4.31375 + 5.76731i 0.273373 + 0.365488i
\(250\) 0 0
\(251\) −1.53757 + 2.66314i −0.0970504 + 0.168096i −0.910462 0.413592i \(-0.864274\pi\)
0.813412 + 0.581688i \(0.197607\pi\)
\(252\) 0 0
\(253\) 13.5385 + 23.4493i 0.851157 + 1.47425i
\(254\) 0 0
\(255\) −1.49978 + 26.7857i −0.0939200 + 1.67738i
\(256\) 0 0
\(257\) −2.47429 + 2.07618i −0.154342 + 0.129508i −0.716689 0.697393i \(-0.754343\pi\)
0.562347 + 0.826902i \(0.309899\pi\)
\(258\) 0 0
\(259\) −16.4098 + 5.97268i −1.01966 + 0.371124i
\(260\) 0 0
\(261\) −6.86544 15.7215i −0.424960 0.973136i
\(262\) 0 0
\(263\) −1.51553 + 8.59501i −0.0934517 + 0.529991i 0.901759 + 0.432239i \(0.142276\pi\)
−0.995211 + 0.0977521i \(0.968835\pi\)
\(264\) 0 0
\(265\) −16.0072 5.82616i −0.983316 0.357898i
\(266\) 0 0
\(267\) 11.3963 12.1338i 0.697440 0.742574i
\(268\) 0 0
\(269\) −8.53979 −0.520680 −0.260340 0.965517i \(-0.583835\pi\)
−0.260340 + 0.965517i \(0.583835\pi\)
\(270\) 0 0
\(271\) 3.54251 0.215192 0.107596 0.994195i \(-0.465685\pi\)
0.107596 + 0.994195i \(0.465685\pi\)
\(272\) 0 0
\(273\) 5.34095 + 17.6954i 0.323249 + 1.07097i
\(274\) 0 0
\(275\) −14.4029 5.24223i −0.868528 0.316118i
\(276\) 0 0
\(277\) 3.16669 17.9592i 0.190268 1.07906i −0.728730 0.684801i \(-0.759889\pi\)
0.918998 0.394262i \(-0.129000\pi\)
\(278\) 0 0
\(279\) −5.64958 + 19.1839i −0.338231 + 1.14851i
\(280\) 0 0
\(281\) 14.8296 5.39755i 0.884663 0.321991i 0.140573 0.990070i \(-0.455106\pi\)
0.744090 + 0.668079i \(0.232883\pi\)
\(282\) 0 0
\(283\) 14.3307 12.0249i 0.851873 0.714806i −0.108328 0.994115i \(-0.534550\pi\)
0.960201 + 0.279309i \(0.0901054\pi\)
\(284\) 0 0
\(285\) 0.217266 0.109726i 0.0128697 0.00649962i
\(286\) 0 0
\(287\) 1.41532 + 2.45141i 0.0835437 + 0.144702i
\(288\) 0 0
\(289\) −6.48111 + 11.2256i −0.381242 + 0.660330i
\(290\) 0 0
\(291\) 29.2392 3.48410i 1.71403 0.204242i
\(292\) 0 0
\(293\) −1.56138 8.85504i −0.0912170 0.517317i −0.995841 0.0911069i \(-0.970959\pi\)
0.904624 0.426210i \(-0.140152\pi\)
\(294\) 0 0
\(295\) −0.0218135 0.0183037i −0.00127003 0.00106568i
\(296\) 0 0
\(297\) −24.9491 8.89098i −1.44769 0.515907i
\(298\) 0 0
\(299\) 8.64588 + 7.25475i 0.500004 + 0.419553i
\(300\) 0 0
\(301\) −8.26987 46.9007i −0.476667 2.70331i
\(302\) 0 0
\(303\) 7.38538 17.2274i 0.424279 0.989690i
\(304\) 0 0
\(305\) −13.7317 + 23.7840i −0.786275 + 1.36187i
\(306\) 0 0
\(307\) −14.9163 25.8358i −0.851319 1.47453i −0.880018 0.474940i \(-0.842470\pi\)
0.0286991 0.999588i \(-0.490864\pi\)
\(308\) 0 0
\(309\) 10.4401 + 6.83308i 0.593919 + 0.388720i
\(310\) 0 0
\(311\) 8.67326 7.27773i 0.491815 0.412682i −0.362861 0.931843i \(-0.618200\pi\)
0.854676 + 0.519161i \(0.173756\pi\)
\(312\) 0 0
\(313\) 8.19289 2.98197i 0.463089 0.168551i −0.0999299 0.994994i \(-0.531862\pi\)
0.563019 + 0.826444i \(0.309640\pi\)
\(314\) 0 0
\(315\) −35.5181 + 23.5897i −2.00122 + 1.32913i
\(316\) 0 0
\(317\) −0.710402 + 4.02889i −0.0399001 + 0.226285i −0.998237 0.0593559i \(-0.981095\pi\)
0.958337 + 0.285641i \(0.0922064\pi\)
\(318\) 0 0
\(319\) 27.3901 + 9.96918i 1.53355 + 0.558167i
\(320\) 0 0
\(321\) 12.6465 + 2.96731i 0.705858 + 0.165619i
\(322\) 0 0
\(323\) 0.271841 0.0151256
\(324\) 0 0
\(325\) −6.38880 −0.354387
\(326\) 0 0
\(327\) −8.85112 2.07679i −0.489468 0.114847i
\(328\) 0 0
\(329\) 9.03521 + 3.28855i 0.498127 + 0.181304i
\(330\) 0 0
\(331\) 2.62145 14.8670i 0.144088 0.817163i −0.824007 0.566579i \(-0.808266\pi\)
0.968095 0.250583i \(-0.0806224\pi\)
\(332\) 0 0
\(333\) −0.653194 10.4098i −0.0357948 0.570454i
\(334\) 0 0
\(335\) 8.31607 3.02680i 0.454355 0.165372i
\(336\) 0 0
\(337\) 16.0967 13.5067i 0.876842 0.735757i −0.0886856 0.996060i \(-0.528267\pi\)
0.965527 + 0.260302i \(0.0838222\pi\)
\(338\) 0 0
\(339\) −12.3169 8.06144i −0.668964 0.437837i
\(340\) 0 0
\(341\) −16.9894 29.4266i −0.920030 1.59354i
\(342\) 0 0
\(343\) −28.1982 + 48.8407i −1.52256 + 2.63715i
\(344\) 0 0
\(345\) −10.2583 + 23.9290i −0.552291 + 1.28830i
\(346\) 0 0
\(347\) −4.39642 24.9333i −0.236012 1.33849i −0.840472 0.541855i \(-0.817722\pi\)
0.604460 0.796636i \(-0.293389\pi\)
\(348\) 0 0
\(349\) −15.9862 13.4140i −0.855720 0.718035i 0.105321 0.994438i \(-0.466413\pi\)
−0.961042 + 0.276404i \(0.910857\pi\)
\(350\) 0 0
\(351\) −11.0398 + 0.0743225i −0.589261 + 0.00396704i
\(352\) 0 0
\(353\) −12.2469 10.2764i −0.651836 0.546955i 0.255792 0.966732i \(-0.417664\pi\)
−0.907627 + 0.419777i \(0.862108\pi\)
\(354\) 0 0
\(355\) −3.18532 18.0649i −0.169059 0.958784i
\(356\) 0 0
\(357\) −47.2856 + 5.63448i −2.50262 + 0.298208i
\(358\) 0 0
\(359\) 13.0281 22.5653i 0.687596 1.19095i −0.285017 0.958523i \(-0.591999\pi\)
0.972613 0.232429i \(-0.0746675\pi\)
\(360\) 0 0
\(361\) 9.49877 + 16.4523i 0.499935 + 0.865913i
\(362\) 0 0
\(363\) 23.1628 11.6980i 1.21573 0.613983i
\(364\) 0 0
\(365\) −13.6843 + 11.4825i −0.716269 + 0.601021i
\(366\) 0 0
\(367\) 8.34015 3.03557i 0.435352 0.158455i −0.115041 0.993361i \(-0.536700\pi\)
0.550393 + 0.834905i \(0.314478\pi\)
\(368\) 0 0
\(369\) −1.64335 + 0.397279i −0.0855493 + 0.0206815i
\(370\) 0 0
\(371\) 5.25060 29.7776i 0.272598 1.54598i
\(372\) 0 0
\(373\) −0.00829968 0.00302084i −0.000429741 0.000156413i 0.341805 0.939771i \(-0.388962\pi\)
−0.342235 + 0.939614i \(0.611184\pi\)
\(374\) 0 0
\(375\) 2.82249 + 9.35136i 0.145753 + 0.482902i
\(376\) 0 0
\(377\) 12.1496 0.625738
\(378\) 0 0
\(379\) 3.59516 0.184671 0.0923354 0.995728i \(-0.470567\pi\)
0.0923354 + 0.995728i \(0.470567\pi\)
\(380\) 0 0
\(381\) −7.61228 + 8.10491i −0.389989 + 0.415227i
\(382\) 0 0
\(383\) 12.5572 + 4.57043i 0.641641 + 0.233538i 0.642290 0.766462i \(-0.277984\pi\)
−0.000649334 1.00000i \(0.500207\pi\)
\(384\) 0 0
\(385\) 12.5800 71.3449i 0.641137 3.63607i
\(386\) 0 0
\(387\) 28.2673 + 3.17544i 1.43691 + 0.161416i
\(388\) 0 0
\(389\) 23.0216 8.37918i 1.16724 0.424841i 0.315563 0.948905i \(-0.397807\pi\)
0.851679 + 0.524063i \(0.175585\pi\)
\(390\) 0 0
\(391\) −22.2744 + 18.6905i −1.12647 + 0.945218i
\(392\) 0 0
\(393\) −1.86445 + 33.2986i −0.0940492 + 1.67969i
\(394\) 0 0
\(395\) 8.25522 + 14.2985i 0.415365 + 0.719434i
\(396\) 0 0
\(397\) 5.00960 8.67688i 0.251425 0.435480i −0.712494 0.701679i \(-0.752434\pi\)
0.963918 + 0.266198i \(0.0857676\pi\)
\(398\) 0 0
\(399\) 0.258778 + 0.345976i 0.0129551 + 0.0173204i
\(400\) 0 0
\(401\) −3.22027 18.2630i −0.160812 0.912013i −0.953278 0.302095i \(-0.902314\pi\)
0.792465 0.609917i \(-0.208797\pi\)
\(402\) 0 0
\(403\) −10.8497 9.10399i −0.540463 0.453502i
\(404\) 0 0
\(405\) −7.52270 24.3305i −0.373806 1.20899i
\(406\) 0 0
\(407\) 13.5757 + 11.3914i 0.672924 + 0.564650i
\(408\) 0 0
\(409\) 1.11206 + 6.30682i 0.0549879 + 0.311852i 0.999879 0.0155318i \(-0.00494413\pi\)
−0.944891 + 0.327384i \(0.893833\pi\)
\(410\) 0 0
\(411\) 2.38503 + 3.18869i 0.117645 + 0.157287i
\(412\) 0 0
\(413\) 0.0252725 0.0437733i 0.00124358 0.00215395i
\(414\) 0 0
\(415\) −5.88306 10.1898i −0.288788 0.500195i
\(416\) 0 0
\(417\) 0.819385 14.6340i 0.0401255 0.716628i
\(418\) 0 0
\(419\) 10.7223 8.99711i 0.523820 0.439538i −0.342141 0.939649i \(-0.611152\pi\)
0.865961 + 0.500111i \(0.166708\pi\)
\(420\) 0 0
\(421\) −3.83960 + 1.39750i −0.187131 + 0.0681100i −0.433886 0.900968i \(-0.642858\pi\)
0.246755 + 0.969078i \(0.420636\pi\)
\(422\) 0 0
\(423\) −3.40871 + 4.62186i −0.165737 + 0.224722i
\(424\) 0 0
\(425\) 2.85817 16.2095i 0.138641 0.786275i
\(426\) 0 0
\(427\) −45.8088 16.6730i −2.21684 0.806864i
\(428\) 0 0
\(429\) 12.8418 13.6728i 0.620008 0.660131i
\(430\) 0 0
\(431\) −28.5009 −1.37284 −0.686419 0.727206i \(-0.740818\pi\)
−0.686419 + 0.727206i \(0.740818\pi\)
\(432\) 0 0
\(433\) −26.9170 −1.29355 −0.646774 0.762682i \(-0.723882\pi\)
−0.646774 + 0.762682i \(0.723882\pi\)
\(434\) 0 0
\(435\) 8.09832 + 26.8310i 0.388285 + 1.28645i
\(436\) 0 0
\(437\) 0.247902 + 0.0902289i 0.0118588 + 0.00431624i
\(438\) 0 0
\(439\) 1.52083 8.62504i 0.0725851 0.411651i −0.926766 0.375639i \(-0.877423\pi\)
0.999351 0.0360121i \(-0.0114655\pi\)
\(440\) 0 0
\(441\) −37.7049 39.6072i −1.79547 1.88606i
\(442\) 0 0
\(443\) −4.86329 + 1.77009i −0.231062 + 0.0840997i −0.454956 0.890514i \(-0.650345\pi\)
0.223894 + 0.974613i \(0.428123\pi\)
\(444\) 0 0
\(445\) −20.8328 + 17.4808i −0.987571 + 0.828670i
\(446\) 0 0
\(447\) −10.3539 + 5.22903i −0.489721 + 0.247325i
\(448\) 0 0
\(449\) −4.15712 7.20034i −0.196187 0.339805i 0.751102 0.660186i \(-0.229523\pi\)
−0.947289 + 0.320381i \(0.896189\pi\)
\(450\) 0 0
\(451\) 1.43630 2.48775i 0.0676328 0.117144i
\(452\) 0 0
\(453\) 4.45946 0.531382i 0.209523 0.0249665i
\(454\) 0 0
\(455\) −5.24368 29.7384i −0.245828 1.39416i
\(456\) 0 0
\(457\) −13.4174 11.2585i −0.627639 0.526652i 0.272555 0.962140i \(-0.412131\pi\)
−0.900194 + 0.435489i \(0.856576\pi\)
\(458\) 0 0
\(459\) 4.75032 28.0431i 0.221726 1.30894i
\(460\) 0 0
\(461\) 11.9588 + 10.0346i 0.556978 + 0.467360i 0.877296 0.479950i \(-0.159345\pi\)
−0.320318 + 0.947310i \(0.603790\pi\)
\(462\) 0 0
\(463\) 6.56692 + 37.2429i 0.305191 + 1.73082i 0.622605 + 0.782536i \(0.286074\pi\)
−0.317415 + 0.948287i \(0.602815\pi\)
\(464\) 0 0
\(465\) 12.8732 30.0285i 0.596980 1.39254i
\(466\) 0 0
\(467\) −6.64253 + 11.5052i −0.307380 + 0.532397i −0.977788 0.209595i \(-0.932786\pi\)
0.670409 + 0.741992i \(0.266119\pi\)
\(468\) 0 0
\(469\) 7.85436 + 13.6041i 0.362680 + 0.628181i
\(470\) 0 0
\(471\) −31.7584 20.7859i −1.46335 0.957764i
\(472\) 0 0
\(473\) −37.0232 + 31.0661i −1.70233 + 1.42842i
\(474\) 0 0
\(475\) −0.140328 + 0.0510752i −0.00643869 + 0.00234349i
\(476\) 0 0
\(477\) 16.1752 + 8.03279i 0.740612 + 0.367796i
\(478\) 0 0
\(479\) 0.495344 2.80924i 0.0226328 0.128357i −0.971398 0.237458i \(-0.923686\pi\)
0.994031 + 0.109100i \(0.0347970\pi\)
\(480\) 0 0
\(481\) 6.94144 + 2.52648i 0.316503 + 0.115198i
\(482\) 0 0
\(483\) −44.9917 10.5566i −2.04719 0.480344i
\(484\) 0 0
\(485\) −48.1062 −2.18439
\(486\) 0 0
\(487\) −6.63845 −0.300817 −0.150409 0.988624i \(-0.548059\pi\)
−0.150409 + 0.988624i \(0.548059\pi\)
\(488\) 0 0
\(489\) 12.6626 + 2.97109i 0.572621 + 0.134357i
\(490\) 0 0
\(491\) 12.9620 + 4.71778i 0.584966 + 0.212910i 0.617514 0.786560i \(-0.288140\pi\)
−0.0325478 + 0.999470i \(0.510362\pi\)
\(492\) 0 0
\(493\) −5.43539 + 30.8256i −0.244798 + 1.38832i
\(494\) 0 0
\(495\) 38.7545 + 19.2459i 1.74189 + 0.865040i
\(496\) 0 0
\(497\) 30.5969 11.1364i 1.37246 0.499534i
\(498\) 0 0
\(499\) −27.7517 + 23.2865i −1.24234 + 1.04245i −0.245000 + 0.969523i \(0.578788\pi\)
−0.997338 + 0.0729224i \(0.976767\pi\)
\(500\) 0 0
\(501\) −3.56432 2.33285i −0.159242 0.104224i
\(502\) 0 0
\(503\) 1.12155 + 1.94258i 0.0500075 + 0.0866155i 0.889946 0.456067i \(-0.150742\pi\)
−0.839938 + 0.542682i \(0.817409\pi\)
\(504\) 0 0
\(505\) −15.3109 + 26.5192i −0.681325 + 1.18009i
\(506\) 0 0
\(507\) −5.79123 + 13.5089i −0.257198 + 0.599950i
\(508\) 0 0
\(509\) 1.18627 + 6.72769i 0.0525806 + 0.298200i 0.999746 0.0225467i \(-0.00717744\pi\)
−0.947165 + 0.320746i \(0.896066\pi\)
\(510\) 0 0
\(511\) −24.2902 20.3819i −1.07454 0.901642i
\(512\) 0 0
\(513\) −0.241892 + 0.0898900i −0.0106798 + 0.00396874i
\(514\) 0 0
\(515\) −15.6154 13.1029i −0.688099 0.577384i
\(516\) 0 0
\(517\) −1.69439 9.60938i −0.0745193 0.422620i
\(518\) 0 0
\(519\) 13.9593 1.66337i 0.612746 0.0730139i
\(520\) 0 0
\(521\) 22.3542 38.7186i 0.979354 1.69629i 0.314607 0.949222i \(-0.398127\pi\)
0.664747 0.747068i \(-0.268539\pi\)
\(522\) 0 0
\(523\) −11.8553 20.5339i −0.518395 0.897886i −0.999772 0.0213722i \(-0.993197\pi\)
0.481377 0.876514i \(-0.340137\pi\)
\(524\) 0 0
\(525\) 23.3508 11.7929i 1.01911 0.514685i
\(526\) 0 0
\(527\) 27.9522 23.4547i 1.21762 1.02170i
\(528\) 0 0
\(529\) −4.90366 + 1.78479i −0.213203 + 0.0775995i
\(530\) 0 0
\(531\) 0.0208157 + 0.0218659i 0.000903325 + 0.000948900i
\(532\) 0 0
\(533\) 0.207922 1.17919i 0.00900612 0.0510762i
\(534\) 0 0
\(535\) −19.9419 7.25826i −0.862164 0.313802i
\(536\) 0 0
\(537\) −6.95843 23.0543i −0.300278 0.994869i
\(538\) 0 0
\(539\) 92.9131 4.00205
\(540\) 0 0
\(541\) 35.1234 1.51007 0.755036 0.655683i \(-0.227619\pi\)
0.755036 + 0.655683i \(0.227619\pi\)
\(542\) 0 0
\(543\) −28.3539 + 30.1888i −1.21678 + 1.29552i
\(544\) 0 0
\(545\) 13.9571 + 5.07997i 0.597856 + 0.217602i
\(546\) 0 0
\(547\) −3.93365 + 22.3088i −0.168191 + 0.953856i 0.777523 + 0.628855i \(0.216476\pi\)
−0.945714 + 0.325001i \(0.894635\pi\)
\(548\) 0 0
\(549\) 17.2823 23.4329i 0.737589 1.00009i
\(550\) 0 0
\(551\) 0.266863 0.0971301i 0.0113687 0.00413788i
\(552\) 0 0
\(553\) −22.4503 + 18.8380i −0.954682 + 0.801073i
\(554\) 0 0
\(555\) −0.952612 + 17.0134i −0.0404361 + 0.722177i
\(556\) 0 0
\(557\) 1.12883 + 1.95519i 0.0478301 + 0.0828442i 0.888949 0.458006i \(-0.151436\pi\)
−0.841119 + 0.540850i \(0.818103\pi\)
\(558\) 0 0
\(559\) −10.0727 + 17.4464i −0.426029 + 0.737903i
\(560\) 0 0
\(561\) 28.9453 + 38.6986i 1.22207 + 1.63386i
\(562\) 0 0
\(563\) −3.24420 18.3988i −0.136727 0.775417i −0.973642 0.228083i \(-0.926754\pi\)
0.836915 0.547333i \(-0.184357\pi\)
\(564\) 0 0
\(565\) 18.4226 + 15.4584i 0.775045 + 0.650340i
\(566\) 0 0
\(567\) 40.2128 20.6497i 1.68878 0.867206i
\(568\) 0 0
\(569\) −30.9156 25.9413i −1.29605 1.08751i −0.990813 0.135238i \(-0.956820\pi\)
−0.305236 0.952277i \(-0.598735\pi\)
\(570\) 0 0
\(571\) 6.10402 + 34.6176i 0.255445 + 1.44870i 0.794927 + 0.606706i \(0.207509\pi\)
−0.539481 + 0.841998i \(0.681380\pi\)
\(572\) 0 0
\(573\) 22.9114 + 30.6316i 0.957137 + 1.27965i
\(574\) 0 0
\(575\) 7.98668 13.8333i 0.333068 0.576890i
\(576\) 0 0
\(577\) −11.8558 20.5349i −0.493565 0.854880i 0.506407 0.862294i \(-0.330973\pi\)
−0.999973 + 0.00741458i \(0.997640\pi\)
\(578\) 0 0
\(579\) −2.40527 + 42.9573i −0.0999595 + 1.78525i
\(580\) 0 0
\(581\) 15.9991 13.4248i 0.663754 0.556956i
\(582\) 0 0
\(583\) −28.8347 + 10.4950i −1.19421 + 0.434658i
\(584\) 0 0
\(585\) 17.9235 + 2.01345i 0.741044 + 0.0832460i
\(586\) 0 0
\(587\) 0.743605 4.21720i 0.0306919 0.174062i −0.965609 0.260000i \(-0.916277\pi\)
0.996301 + 0.0859376i \(0.0273886\pi\)
\(588\) 0 0
\(589\) −0.311092 0.113228i −0.0128183 0.00466549i
\(590\) 0 0
\(591\) 4.36703 4.64964i 0.179636 0.191261i
\(592\) 0 0
\(593\) −10.5439 −0.432986 −0.216493 0.976284i \(-0.569462\pi\)
−0.216493 + 0.976284i \(0.569462\pi\)
\(594\) 0 0
\(595\) 77.7972 3.18938
\(596\) 0 0
\(597\) −12.7541 42.2563i −0.521991 1.72944i
\(598\) 0 0
\(599\) 44.6336 + 16.2453i 1.82368 + 0.663766i 0.994494 + 0.104795i \(0.0334186\pi\)
0.829187 + 0.558971i \(0.188804\pi\)
\(600\) 0 0
\(601\) −3.55031 + 20.1348i −0.144820 + 0.821315i 0.822691 + 0.568488i \(0.192472\pi\)
−0.967511 + 0.252827i \(0.918640\pi\)
\(602\) 0 0
\(603\) −9.11980 + 2.20471i −0.371387 + 0.0897826i
\(604\) 0 0
\(605\) −39.8366 + 14.4993i −1.61959 + 0.589482i
\(606\) 0 0
\(607\) 15.1465 12.7094i 0.614777 0.515859i −0.281380 0.959597i \(-0.590792\pi\)
0.896157 + 0.443737i \(0.146348\pi\)
\(608\) 0 0
\(609\) −44.4064 + 22.4266i −1.79944 + 0.908774i
\(610\) 0 0
\(611\) −2.03362 3.52233i −0.0822713 0.142498i
\(612\) 0 0
\(613\) −3.03815 + 5.26222i −0.122710 + 0.212539i −0.920835 0.389952i \(-0.872492\pi\)
0.798126 + 0.602491i \(0.205825\pi\)
\(614\) 0 0
\(615\) 2.74268 0.326814i 0.110596 0.0131784i
\(616\) 0 0
\(617\) 1.60518 + 9.10342i 0.0646221 + 0.366490i 0.999920 + 0.0126294i \(0.00402018\pi\)
−0.935298 + 0.353861i \(0.884869\pi\)
\(618\) 0 0
\(619\) −11.4567 9.61333i −0.460484 0.386392i 0.382825 0.923821i \(-0.374951\pi\)
−0.843309 + 0.537429i \(0.819396\pi\)
\(620\) 0 0
\(621\) 13.6400 23.9968i 0.547354 0.962959i
\(622\) 0 0
\(623\) −36.9791 31.0292i −1.48154 1.24316i
\(624\) 0 0
\(625\) −5.38187 30.5221i −0.215275 1.22088i
\(626\) 0 0
\(627\) 0.172758 0.402983i 0.00689931 0.0160936i
\(628\) 0 0
\(629\) −9.51551 + 16.4813i −0.379408 + 0.657154i
\(630\) 0 0
\(631\) 10.4381 + 18.0793i 0.415534 + 0.719726i 0.995484 0.0949256i \(-0.0302613\pi\)
−0.579950 + 0.814652i \(0.696928\pi\)
\(632\) 0 0
\(633\) 3.02605 + 1.98055i 0.120275 + 0.0787198i
\(634\) 0 0
\(635\) 13.9156 11.6765i 0.552223 0.463370i
\(636\) 0 0
\(637\) 36.3930 13.2460i 1.44194 0.524824i
\(638\) 0 0
\(639\) 1.21791 + 19.4096i 0.0481799 + 0.767833i
\(640\) 0 0
\(641\) −2.71244 + 15.3830i −0.107135 + 0.607592i 0.883211 + 0.468975i \(0.155377\pi\)
−0.990346 + 0.138617i \(0.955734\pi\)
\(642\) 0 0
\(643\) −18.6682 6.79466i −0.736201 0.267955i −0.0534137 0.998572i \(-0.517010\pi\)
−0.682787 + 0.730617i \(0.739232\pi\)
\(644\) 0 0
\(645\) −45.2421 10.6154i −1.78141 0.417981i
\(646\) 0 0
\(647\) 15.4741 0.608349 0.304174 0.952616i \(-0.401619\pi\)
0.304174 + 0.952616i \(0.401619\pi\)
\(648\) 0 0
\(649\) −0.0512944 −0.00201348
\(650\) 0 0
\(651\) 56.4600 + 13.2475i 2.21284 + 0.519212i
\(652\) 0 0
\(653\) 15.9016 + 5.78772i 0.622279 + 0.226491i 0.633867 0.773442i \(-0.281467\pi\)
−0.0115885 + 0.999933i \(0.503689\pi\)
\(654\) 0 0
\(655\) 9.46126 53.6575i 0.369682 2.09657i
\(656\) 0 0
\(657\) 15.7764 10.4780i 0.615495 0.408787i
\(658\) 0 0
\(659\) 14.2658 5.19231i 0.555715 0.202264i −0.0488689 0.998805i \(-0.515562\pi\)
0.604584 + 0.796541i \(0.293339\pi\)
\(660\) 0 0
\(661\) −28.6261 + 24.0201i −1.11343 + 0.934276i −0.998254 0.0590700i \(-0.981186\pi\)
−0.115172 + 0.993346i \(0.536742\pi\)
\(662\) 0 0
\(663\) 16.8545 + 11.0313i 0.654575 + 0.428420i
\(664\) 0 0
\(665\) −0.352919 0.611274i −0.0136856 0.0237042i
\(666\) 0 0
\(667\) −15.1883 + 26.3069i −0.588094 + 1.01861i
\(668\) 0 0
\(669\) −0.585920 + 1.36674i −0.0226530 + 0.0528413i
\(670\) 0 0
\(671\) 8.59061 + 48.7198i 0.331637 + 1.88081i
\(672\) 0 0
\(673\) 21.5116 + 18.0504i 0.829211 + 0.695791i 0.955110 0.296252i \(-0.0957369\pi\)
−0.125898 + 0.992043i \(0.540181\pi\)
\(674\) 0 0
\(675\) 2.81673 + 15.3687i 0.108416 + 0.591543i
\(676\) 0 0
\(677\) −10.7694 9.03664i −0.413903 0.347306i 0.411935 0.911213i \(-0.364853\pi\)
−0.825838 + 0.563907i \(0.809298\pi\)
\(678\) 0 0
\(679\) −14.8279 84.0932i −0.569043 3.22720i
\(680\) 0 0
\(681\) 12.4554 1.48417i 0.477291 0.0568734i
\(682\) 0 0
\(683\) −2.49945 + 4.32918i −0.0956389 + 0.165651i −0.909875 0.414882i \(-0.863823\pi\)
0.814236 + 0.580534i \(0.197156\pi\)
\(684\) 0 0
\(685\) −3.25269 5.63382i −0.124279 0.215257i
\(686\) 0 0
\(687\) −24.7746 + 12.5120i −0.945209 + 0.477361i
\(688\) 0 0
\(689\) −9.79804 + 8.22153i −0.373276 + 0.313215i
\(690\) 0 0
\(691\) −19.1044 + 6.95345i −0.726767 + 0.264522i −0.678796 0.734327i \(-0.737498\pi\)
−0.0479713 + 0.998849i \(0.515276\pi\)
\(692\) 0 0
\(693\) −21.6979 + 73.6780i −0.824235 + 2.79879i
\(694\) 0 0
\(695\) −4.15801 + 23.5813i −0.157722 + 0.894488i
\(696\) 0 0
\(697\) 2.89878 + 1.05507i 0.109799 + 0.0399636i
\(698\) 0 0
\(699\) −11.8537 39.2733i −0.448350 1.48545i
\(700\) 0 0
\(701\) −0.985890 −0.0372366 −0.0186183 0.999827i \(-0.505927\pi\)
−0.0186183 + 0.999827i \(0.505927\pi\)
\(702\) 0 0
\(703\) 0.172665 0.00651217
\(704\) 0 0
\(705\) 6.42313 6.83880i 0.241909 0.257564i
\(706\) 0 0
\(707\) −51.0769 18.5905i −1.92094 0.699166i
\(708\) 0 0
\(709\) 0.153891 0.872759i 0.00577949 0.0327771i −0.981782 0.190013i \(-0.939147\pi\)
0.987561 + 0.157236i \(0.0502582\pi\)
\(710\) 0 0
\(711\) −7.00518 16.0415i −0.262715 0.601603i
\(712\) 0 0
\(713\) 33.2757 12.1114i 1.24618 0.453574i
\(714\) 0 0
\(715\) −23.4753 + 19.6981i −0.877927 + 0.736668i
\(716\) 0 0
\(717\) −1.30636 + 23.3311i −0.0487867 + 0.871316i
\(718\) 0 0
\(719\) −14.9029 25.8125i −0.555783 0.962644i −0.997842 0.0656580i \(-0.979085\pi\)
0.442060 0.896986i \(-0.354248\pi\)
\(720\) 0 0
\(721\) 18.0917 31.3357i 0.673770 1.16700i
\(722\) 0 0
\(723\) −2.33474 3.12145i −0.0868297 0.116088i
\(724\) 0 0
\(725\) −2.98589 16.9338i −0.110893 0.628907i
\(726\) 0 0
\(727\) 8.79469 + 7.37962i 0.326177 + 0.273695i 0.791140 0.611635i \(-0.209488\pi\)
−0.464963 + 0.885330i \(0.653932\pi\)
\(728\) 0 0
\(729\) 5.04608 + 26.5243i 0.186892 + 0.982381i
\(730\) 0 0
\(731\) −39.7582 33.3611i −1.47051 1.23390i
\(732\) 0 0
\(733\) −6.11811 34.6975i −0.225978 1.28158i −0.860807 0.508931i \(-0.830041\pi\)
0.634830 0.772652i \(-0.281070\pi\)
\(734\) 0 0
\(735\) 53.5098 + 71.5405i 1.97374 + 2.63881i
\(736\) 0 0
\(737\) 7.97080 13.8058i 0.293608 0.508544i
\(738\) 0 0
\(739\) −11.1188 19.2584i −0.409013 0.708432i 0.585766 0.810480i \(-0.300794\pi\)
−0.994779 + 0.102048i \(0.967460\pi\)
\(740\) 0 0
\(741\) 0.0102170 0.182473i 0.000375332 0.00670331i
\(742\) 0 0
\(743\) −2.25293 + 1.89044i −0.0826521 + 0.0693534i −0.683178 0.730252i \(-0.739403\pi\)
0.600526 + 0.799606i \(0.294958\pi\)
\(744\) 0 0
\(745\) 17.8071 6.48126i 0.652402 0.237455i
\(746\) 0 0
\(747\) 4.99222 + 11.4319i 0.182656 + 0.418272i
\(748\) 0 0
\(749\) 6.54123 37.0972i 0.239011 1.35550i
\(750\) 0 0
\(751\) −7.83959 2.85338i −0.286071 0.104121i 0.194999 0.980803i \(-0.437530\pi\)
−0.481070 + 0.876682i \(0.659752\pi\)
\(752\) 0 0
\(753\) −3.64642 + 3.88239i −0.132883 + 0.141482i
\(754\) 0 0
\(755\) −7.33698 −0.267020
\(756\) 0 0
\(757\) 23.4397 0.851930 0.425965 0.904740i \(-0.359935\pi\)
0.425965 + 0.904740i \(0.359935\pi\)
\(758\) 0 0
\(759\) 13.5515 + 44.8981i 0.491888 + 1.62970i
\(760\) 0 0
\(761\) −13.0388 4.74575i −0.472657 0.172033i 0.0946985 0.995506i \(-0.469811\pi\)
−0.567356 + 0.823473i \(0.692034\pi\)
\(762\) 0 0
\(763\) −4.57813 + 25.9638i −0.165739 + 0.939954i
\(764\) 0 0
\(765\) −13.1269 + 44.5741i −0.474604 + 1.61158i
\(766\) 0 0
\(767\) −0.0200914 + 0.00731269i −0.000725460 + 0.000264046i
\(768\) 0 0
\(769\) −31.8412 + 26.7179i −1.14822 + 0.963473i −0.999677 0.0254329i \(-0.991904\pi\)
−0.148546 + 0.988906i \(0.547459\pi\)
\(770\) 0 0
\(771\) −4.99374 + 2.52200i −0.179845 + 0.0908275i
\(772\) 0 0
\(773\) 13.0208 + 22.5527i 0.468326 + 0.811165i 0.999345 0.0361955i \(-0.0115239\pi\)
−0.531019 + 0.847360i \(0.678191\pi\)
\(774\) 0 0
\(775\) −10.0225 + 17.3595i −0.360018 + 0.623570i
\(776\) 0 0
\(777\) −30.0342 + 3.57884i −1.07747 + 0.128390i
\(778\) 0 0
\(779\) −0.00486005 0.0275627i −0.000174129 0.000987536i
\(780\) 0 0
\(781\) −25.3126 21.2398i −0.905757 0.760020i
\(782\) 0 0
\(783\) −5.35660 29.2268i −0.191429 1.04448i
\(784\) 0 0
\(785\) 47.5015 + 39.8585i 1.69540 + 1.42261i
\(786\) 0 0
\(787\) −3.95947 22.4553i −0.141140 0.800444i −0.970386 0.241560i \(-0.922341\pi\)
0.829246 0.558884i \(-0.188770\pi\)
\(788\) 0 0
\(789\) −5.95624 + 13.8938i −0.212048 + 0.494631i
\(790\) 0 0
\(791\) −21.3440 + 36.9689i −0.758905 + 1.31446i
\(792\) 0 0
\(793\) 10.3105 + 17.8583i 0.366136 + 0.634166i
\(794\) 0 0
\(795\) −24.6871 16.1577i −0.875562 0.573056i
\(796\) 0 0
\(797\) −33.6971 + 28.2752i −1.19361 + 1.00156i −0.193823 + 0.981037i \(0.562089\pi\)
−0.999789 + 0.0205228i \(0.993467\pi\)
\(798\) 0 0
\(799\) 9.84652 3.58384i 0.348345 0.126787i
\(800\) 0 0
\(801\) 24.0178 15.9516i 0.848627 0.563623i
\(802\) 0 0
\(803\) −5.58777 + 31.6898i −0.197188 + 1.11831i
\(804\) 0 0
\(805\) 70.9462 + 25.8223i 2.50052 + 0.910116i
\(806\) 0 0
\(807\) −14.4003 3.37881i −0.506913 0.118940i
\(808\) 0 0
\(809\) 35.8016 1.25872 0.629358 0.777115i \(-0.283318\pi\)
0.629358 + 0.777115i \(0.283318\pi\)
\(810\) 0 0
\(811\) 48.3005 1.69606 0.848031 0.529947i \(-0.177788\pi\)
0.848031 + 0.529947i \(0.177788\pi\)
\(812\) 0 0
\(813\) 5.97358 + 1.40162i 0.209503 + 0.0491568i
\(814\) 0 0
\(815\) −19.9673 7.26749i −0.699423 0.254569i
\(816\) 0 0
\(817\) −0.0817681 + 0.463730i −0.00286070 + 0.0162239i
\(818\) 0 0
\(819\) 2.00493 + 31.9521i 0.0700579 + 1.11650i
\(820\) 0 0
\(821\) −0.856294 + 0.311666i −0.0298849 + 0.0108772i −0.356919 0.934135i \(-0.616173\pi\)
0.327034 + 0.945012i \(0.393951\pi\)
\(822\) 0 0
\(823\) 14.2399 11.9487i 0.496371 0.416505i −0.359932 0.932978i \(-0.617200\pi\)
0.856303 + 0.516474i \(0.172756\pi\)
\(824\) 0 0
\(825\) −22.2129 14.5383i −0.773353 0.506160i
\(826\) 0 0
\(827\) 15.6029 + 27.0251i 0.542567 + 0.939754i 0.998756 + 0.0498708i \(0.0158809\pi\)
−0.456189 + 0.889883i \(0.650786\pi\)
\(828\) 0 0
\(829\) 16.5336 28.6370i 0.574235 0.994605i −0.421889 0.906648i \(-0.638633\pi\)
0.996124 0.0879573i \(-0.0280339\pi\)
\(830\) 0 0
\(831\) 12.4455 29.0309i 0.431730 1.00707i
\(832\) 0 0
\(833\) 17.3261 + 98.2610i 0.600313 + 3.40454i
\(834\) 0 0
\(835\) 5.33120 + 4.47340i 0.184494 + 0.154809i
\(836\) 0 0
\(837\) −17.1168 + 30.1136i −0.591644 + 1.04088i
\(838\) 0 0
\(839\) 14.4443 + 12.1202i 0.498671 + 0.418435i 0.857122 0.515114i \(-0.172250\pi\)
−0.358451 + 0.933549i \(0.616695\pi\)
\(840\) 0 0
\(841\) 0.642495 + 3.64377i 0.0221550 + 0.125647i
\(842\) 0 0
\(843\) 27.1421 3.23422i 0.934825 0.111392i
\(844\) 0 0
\(845\) 12.0060 20.7950i 0.413019 0.715370i
\(846\) 0 0
\(847\) −37.6249 65.1682i −1.29281 2.23920i
\(848\) 0 0
\(849\) 28.9230 14.6070i 0.992634 0.501312i
\(850\) 0 0
\(851\) −14.1480 + 11.8716i −0.484987 + 0.406952i
\(852\) 0 0
\(853\) 4.86449 1.77053i 0.166557 0.0606218i −0.257396 0.966306i \(-0.582864\pi\)
0.423953 + 0.905684i \(0.360642\pi\)
\(854\) 0 0
\(855\) 0.409779 0.0990640i 0.0140142 0.00338792i
\(856\) 0 0
\(857\) −9.45910 + 53.6452i −0.323117 + 1.83249i 0.199474 + 0.979903i \(0.436077\pi\)
−0.522591 + 0.852583i \(0.675035\pi\)
\(858\) 0 0
\(859\) 20.8775 + 7.59880i 0.712332 + 0.259268i 0.672667 0.739945i \(-0.265149\pi\)
0.0396651 + 0.999213i \(0.487371\pi\)
\(860\) 0 0
\(861\) 1.41668 + 4.69368i 0.0482803 + 0.159960i
\(862\) 0 0
\(863\) −14.3277 −0.487722 −0.243861 0.969810i \(-0.578414\pi\)
−0.243861 + 0.969810i \(0.578414\pi\)
\(864\) 0 0
\(865\) −22.9667 −0.780893
\(866\) 0 0
\(867\) −15.3703 + 16.3650i −0.522002 + 0.555783i
\(868\) 0 0
\(869\) 27.9476 + 10.1721i 0.948057 + 0.345064i
\(870\) 0 0
\(871\) 1.15387 6.54393i 0.0390974 0.221732i
\(872\) 0 0
\(873\) 50.6833 + 5.69357i 1.71537 + 0.192698i
\(874\) 0 0
\(875\) 26.6180 9.68815i 0.899851 0.327519i
\(876\) 0 0
\(877\) −26.9118 + 22.5817i −0.908746 + 0.762529i −0.971880 0.235476i \(-0.924335\pi\)
0.0631337 + 0.998005i \(0.479891\pi\)
\(878\) 0 0
\(879\) 0.870655 15.5496i 0.0293665 0.524476i
\(880\) 0 0
\(881\) 23.9428 + 41.4702i 0.806654 + 1.39717i 0.915169 + 0.403071i \(0.132057\pi\)
−0.108515 + 0.994095i \(0.534610\pi\)
\(882\) 0 0
\(883\) 9.75528 16.8966i 0.328291 0.568617i −0.653882 0.756597i \(-0.726861\pi\)
0.982173 + 0.187980i \(0.0601939\pi\)
\(884\) 0 0
\(885\) −0.0295411 0.0394953i −0.000993014 0.00132762i
\(886\) 0 0
\(887\) −9.94927 56.4251i −0.334064 1.89457i −0.436285 0.899808i \(-0.643706\pi\)
0.102221 0.994762i \(-0.467405\pi\)
\(888\) 0 0
\(889\) 24.7007 + 20.7264i 0.828435 + 0.695140i
\(890\) 0 0
\(891\) −38.5528 24.8637i −1.29157 0.832965i
\(892\) 0 0
\(893\) −0.0728269 0.0611090i −0.00243706 0.00204494i
\(894\) 0 0
\(895\) 6.83170 + 38.7445i 0.228359 + 1.29509i
\(896\) 0 0
\(897\) 11.7088 + 15.6542i 0.390945 + 0.522677i
\(898\) 0 0
\(899\) 19.0598 33.0126i 0.635681 1.10103i
\(900\) 0 0
\(901\) −16.4760 28.5374i −0.548897 0.950717i
\(902\) 0 0
\(903\) 4.61143 82.3586i 0.153459 2.74072i
\(904\) 0 0
\(905\) 51.8320 43.4922i 1.72295 1.44573i
\(906\) 0 0
\(907\) −3.65200 + 1.32922i −0.121263 + 0.0441360i −0.401939 0.915667i \(-0.631664\pi\)
0.280676 + 0.959803i \(0.409441\pi\)
\(908\) 0 0
\(909\) 19.2698 26.1278i 0.639137 0.866604i
\(910\) 0 0
\(911\) 4.62455 26.2271i 0.153218 0.868943i −0.807179 0.590307i \(-0.799007\pi\)
0.960397 0.278636i \(-0.0898823\pi\)
\(912\) 0 0
\(913\) −19.9167 7.24910i −0.659148 0.239910i
\(914\) 0 0
\(915\) −32.5654 + 34.6729i −1.07658 + 1.14625i
\(916\) 0 0
\(917\) 96.7135 3.19376
\(918\) 0 0
\(919\) −32.4014 −1.06882 −0.534412 0.845224i \(-0.679467\pi\)
−0.534412 + 0.845224i \(0.679467\pi\)
\(920\) 0 0
\(921\) −14.9306 49.4675i −0.491981 1.63001i
\(922\) 0 0
\(923\) −12.9427 4.71075i −0.426013 0.155056i
\(924\) 0 0
\(925\) 1.81541 10.2957i 0.0596904 0.338521i
\(926\) 0 0
\(927\) 14.9012 + 15.6530i 0.489420 + 0.514112i
\(928\) 0 0
\(929\) −51.0474 + 18.5797i −1.67481 + 0.609581i −0.992584 0.121561i \(-0.961210\pi\)
−0.682226 + 0.731142i \(0.738988\pi\)
\(930\) 0 0
\(931\) 0.693466 0.581887i 0.0227274 0.0190706i
\(932\) 0 0
\(933\) 17.5048 8.84049i 0.573082 0.289425i
\(934\) 0 0
\(935\) −39.4753 68.3732i −1.29098 2.23604i
\(936\) 0 0
\(937\) −12.9856 + 22.4917i −0.424221 + 0.734773i −0.996347 0.0853929i \(-0.972785\pi\)
0.572126 + 0.820166i \(0.306119\pi\)
\(938\) 0 0
\(939\) 14.9951 1.78680i 0.489348 0.0583100i
\(940\) 0 0
\(941\) 0.735542 + 4.17147i 0.0239780 + 0.135986i 0.994447 0.105241i \(-0.0335614\pi\)
−0.970469 + 0.241227i \(0.922450\pi\)
\(942\) 0 0
\(943\) 2.29330 + 1.92431i 0.0746803 + 0.0626642i
\(944\) 0 0
\(945\) −69.2261 + 25.7253i −2.25192 + 0.836844i
\(946\) 0 0
\(947\) 35.4120 + 29.7142i 1.15073 + 0.965581i 0.999737 0.0229533i \(-0.00730689\pi\)
0.150998 + 0.988534i \(0.451751\pi\)
\(948\) 0 0
\(949\) 2.32913 + 13.2092i 0.0756068 + 0.428787i
\(950\) 0 0
\(951\) −2.79197 + 6.51266i −0.0905358 + 0.211188i
\(952\) 0 0
\(953\) −23.4066 + 40.5414i −0.758213 + 1.31326i 0.185548 + 0.982635i \(0.440594\pi\)
−0.943761 + 0.330628i \(0.892739\pi\)
\(954\) 0 0
\(955\) −31.2463 54.1203i −1.01111 1.75129i
\(956\) 0 0
\(957\) 42.2423 + 27.6476i 1.36550 + 0.893721i
\(958\) 0 0
\(959\) 8.84575 7.42247i 0.285644 0.239684i
\(960\) 0 0
\(961\) −12.6272 + 4.59592i −0.407328 + 0.148255i
\(962\) 0 0
\(963\) 20.1512 + 10.0073i 0.649362 + 0.322481i
\(964\) 0 0
\(965\) 12.2057 69.2217i 0.392914 2.22833i
\(966\) 0 0
\(967\) 41.3709 + 15.0578i 1.33040 + 0.484226i 0.906776 0.421613i \(-0.138536\pi\)
0.423623 + 0.905839i \(0.360758\pi\)
\(968\) 0 0
\(969\) 0.458394 + 0.107555i 0.0147257 + 0.00345518i
\(970\) 0 0
\(971\) −37.4460 −1.20170 −0.600849 0.799363i \(-0.705171\pi\)
−0.600849 + 0.799363i \(0.705171\pi\)
\(972\) 0 0
\(973\) −42.5034 −1.36260
\(974\) 0 0
\(975\) −10.7732 2.52777i −0.345017 0.0809533i
\(976\) 0 0
\(977\) −47.5171 17.2948i −1.52021 0.553310i −0.559006 0.829163i \(-0.688817\pi\)
−0.961200 + 0.275854i \(0.911040\pi\)
\(978\) 0 0
\(979\) −8.50676 + 48.2442i −0.271877 + 1.54189i
\(980\) 0 0
\(981\) −14.1036 7.00398i −0.450292 0.223620i
\(982\) 0 0
\(983\) 41.7808 15.2070i 1.33260 0.485027i 0.425126 0.905134i \(-0.360230\pi\)
0.907474 + 0.420108i \(0.138008\pi\)
\(984\) 0 0
\(985\) −7.98311 + 6.69863i −0.254363 + 0.213436i
\(986\) 0 0
\(987\) 13.9345 + 9.12016i 0.443541 + 0.290298i
\(988\) 0 0
\(989\) −25.1838 43.6196i −0.800799 1.38702i
\(990\) 0 0
\(991\) −24.6074 + 42.6213i −0.781680 + 1.35391i 0.149282 + 0.988795i \(0.452304\pi\)
−0.930962 + 0.365115i \(0.881030\pi\)
\(992\) 0 0
\(993\) 10.3026 24.0323i 0.326944 0.762643i
\(994\) 0 0
\(995\) 12.5218 + 71.0149i 0.396969 + 2.25132i
\(996\) 0 0
\(997\) 32.3050 + 27.1071i 1.02311 + 0.858490i 0.990015 0.140962i \(-0.0450197\pi\)
0.0330934 + 0.999452i \(0.489464\pi\)
\(998\) 0 0
\(999\) 3.01725 17.8120i 0.0954615 0.563548i
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 864.2.y.c.97.9 yes 54
4.3 odd 2 864.2.y.b.97.1 54
27.22 even 9 inner 864.2.y.c.481.9 yes 54
108.103 odd 18 864.2.y.b.481.1 yes 54
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.y.b.97.1 54 4.3 odd 2
864.2.y.b.481.1 yes 54 108.103 odd 18
864.2.y.c.97.9 yes 54 1.1 even 1 trivial
864.2.y.c.481.9 yes 54 27.22 even 9 inner