Properties

Label 432.3.bc.b.65.4
Level $432$
Weight $3$
Character 432.65
Analytic conductor $11.771$
Analytic rank $0$
Dimension $36$
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] = [432,3,Mod(65,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 432.bc (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.7711474204\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 65.4
Character \(\chi\) \(=\) 432.65
Dual form 432.3.bc.b.113.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.727525 - 2.91045i) q^{3} +(0.0686711 - 0.188672i) q^{5} +(1.47862 - 8.38565i) q^{7} +(-7.94141 - 4.23485i) q^{9} +O(q^{10})\) \(q+(0.727525 - 2.91045i) q^{3} +(0.0686711 - 0.188672i) q^{5} +(1.47862 - 8.38565i) q^{7} +(-7.94141 - 4.23485i) q^{9} +(-2.58315 - 7.09715i) q^{11} +(-12.5592 + 10.5384i) q^{13} +(-0.499161 - 0.337127i) q^{15} +(5.21882 - 3.01309i) q^{17} +(0.189946 - 0.328995i) q^{19} +(-23.3303 - 10.4042i) q^{21} +(-27.6819 + 4.88107i) q^{23} +(19.1202 + 16.0438i) q^{25} +(-18.1029 + 20.0321i) q^{27} +(26.6332 - 31.7402i) q^{29} +(-2.35612 - 13.3622i) q^{31} +(-22.5352 + 2.35477i) q^{33} +(-1.48060 - 0.854825i) q^{35} +(-2.26190 - 3.91773i) q^{37} +(21.5344 + 44.2199i) q^{39} +(-49.3383 - 58.7991i) q^{41} +(-1.63966 + 0.596788i) q^{43} +(-1.34434 + 1.20751i) q^{45} +(-75.3795 - 13.2914i) q^{47} +(-22.0879 - 8.03934i) q^{49} +(-4.97261 - 17.3812i) q^{51} +85.8739i q^{53} -1.51642 q^{55} +(-0.819334 - 0.792179i) q^{57} +(-6.23410 + 17.1281i) q^{59} +(-6.51665 + 36.9578i) q^{61} +(-47.2543 + 60.3322i) q^{63} +(1.12585 + 3.09326i) q^{65} +(53.9204 - 45.2446i) q^{67} +(-5.93319 + 84.1179i) q^{69} +(38.9179 - 22.4692i) q^{71} +(51.2495 - 88.7667i) q^{73} +(60.6050 - 43.9762i) q^{75} +(-63.3337 + 11.1674i) q^{77} +(-64.7058 - 54.2946i) q^{79} +(45.1321 + 67.2614i) q^{81} +(44.2042 - 52.6805i) q^{83} +(-0.210104 - 1.19156i) q^{85} +(-73.0020 - 100.606i) q^{87} +(-119.245 - 68.8461i) q^{89} +(69.8014 + 120.900i) q^{91} +(-40.6042 - 2.86398i) q^{93} +(-0.0490285 - 0.0584299i) q^{95} +(112.916 - 41.0979i) q^{97} +(-9.54147 + 67.3007i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 9 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q - 9 q^{5} + 6 q^{9} - 36 q^{11} - 45 q^{15} + 42 q^{21} + 18 q^{23} - 9 q^{25} - 18 q^{29} - 45 q^{31} - 153 q^{33} + 243 q^{35} + 123 q^{39} - 198 q^{41} - 90 q^{43} - 333 q^{45} + 243 q^{47} + 72 q^{49} + 99 q^{51} + 243 q^{57} - 252 q^{59} - 144 q^{61} - 381 q^{63} + 747 q^{65} - 108 q^{67} + 585 q^{69} - 324 q^{71} - 63 q^{73} - 597 q^{75} + 495 q^{77} - 36 q^{79} - 54 q^{81} + 27 q^{83} - 180 q^{85} + 441 q^{87} - 567 q^{89} - 99 q^{91} - 699 q^{93} + 1044 q^{95} - 216 q^{97} + 945 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{13}{18}\right)\)

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
\(3\) 0.727525 2.91045i 0.242508 0.970149i
\(4\) 0 0
\(5\) 0.0686711 0.188672i 0.0137342 0.0377344i −0.932636 0.360818i \(-0.882497\pi\)
0.946370 + 0.323084i \(0.104720\pi\)
\(6\) 0 0
\(7\) 1.47862 8.38565i 0.211231 1.19795i −0.676098 0.736812i \(-0.736330\pi\)
0.887328 0.461138i \(-0.152559\pi\)
\(8\) 0 0
\(9\) −7.94141 4.23485i −0.882379 0.470539i
\(10\) 0 0
\(11\) −2.58315 7.09715i −0.234832 0.645195i −0.999999 0.00136376i \(-0.999566\pi\)
0.765167 0.643832i \(-0.222656\pi\)
\(12\) 0 0
\(13\) −12.5592 + 10.5384i −0.966094 + 0.810649i −0.981934 0.189226i \(-0.939402\pi\)
0.0158399 + 0.999875i \(0.494958\pi\)
\(14\) 0 0
\(15\) −0.499161 0.337127i −0.0332774 0.0224752i
\(16\) 0 0
\(17\) 5.21882 3.01309i 0.306990 0.177241i −0.338589 0.940934i \(-0.609950\pi\)
0.645579 + 0.763694i \(0.276616\pi\)
\(18\) 0 0
\(19\) 0.189946 0.328995i 0.00999714 0.0173155i −0.860984 0.508633i \(-0.830151\pi\)
0.870981 + 0.491317i \(0.163484\pi\)
\(20\) 0 0
\(21\) −23.3303 10.4042i −1.11097 0.495438i
\(22\) 0 0
\(23\) −27.6819 + 4.88107i −1.20356 + 0.212220i −0.739238 0.673445i \(-0.764814\pi\)
−0.464324 + 0.885665i \(0.653703\pi\)
\(24\) 0 0
\(25\) 19.1202 + 16.0438i 0.764809 + 0.641751i
\(26\) 0 0
\(27\) −18.1029 + 20.0321i −0.670477 + 0.741930i
\(28\) 0 0
\(29\) 26.6332 31.7402i 0.918387 1.09449i −0.0768538 0.997042i \(-0.524487\pi\)
0.995241 0.0974484i \(-0.0310681\pi\)
\(30\) 0 0
\(31\) −2.35612 13.3622i −0.0760038 0.431039i −0.998938 0.0460807i \(-0.985327\pi\)
0.922934 0.384959i \(-0.125784\pi\)
\(32\) 0 0
\(33\) −22.5352 + 2.35477i −0.682885 + 0.0713568i
\(34\) 0 0
\(35\) −1.48060 0.854825i −0.0423029 0.0244236i
\(36\) 0 0
\(37\) −2.26190 3.91773i −0.0611325 0.105885i 0.833839 0.552007i \(-0.186138\pi\)
−0.894972 + 0.446122i \(0.852805\pi\)
\(38\) 0 0
\(39\) 21.5344 + 44.2199i 0.552165 + 1.13384i
\(40\) 0 0
\(41\) −49.3383 58.7991i −1.20337 1.43412i −0.871217 0.490899i \(-0.836668\pi\)
−0.332156 0.943225i \(-0.607776\pi\)
\(42\) 0 0
\(43\) −1.63966 + 0.596788i −0.0381317 + 0.0138788i −0.361016 0.932560i \(-0.617570\pi\)
0.322884 + 0.946439i \(0.395348\pi\)
\(44\) 0 0
\(45\) −1.34434 + 1.20751i −0.0298743 + 0.0268336i
\(46\) 0 0
\(47\) −75.3795 13.2914i −1.60382 0.282797i −0.701112 0.713051i \(-0.747313\pi\)
−0.902708 + 0.430255i \(0.858424\pi\)
\(48\) 0 0
\(49\) −22.0879 8.03934i −0.450773 0.164068i
\(50\) 0 0
\(51\) −4.97261 17.3812i −0.0975022 0.340808i
\(52\) 0 0
\(53\) 85.8739i 1.62026i 0.586249 + 0.810131i \(0.300604\pi\)
−0.586249 + 0.810131i \(0.699396\pi\)
\(54\) 0 0
\(55\) −1.51642 −0.0275713
\(56\) 0 0
\(57\) −0.819334 0.792179i −0.0143743 0.0138979i
\(58\) 0 0
\(59\) −6.23410 + 17.1281i −0.105663 + 0.290306i −0.981246 0.192762i \(-0.938255\pi\)
0.875583 + 0.483068i \(0.160478\pi\)
\(60\) 0 0
\(61\) −6.51665 + 36.9578i −0.106830 + 0.605865i 0.883643 + 0.468161i \(0.155083\pi\)
−0.990473 + 0.137704i \(0.956028\pi\)
\(62\) 0 0
\(63\) −47.2543 + 60.3322i −0.750068 + 0.957654i
\(64\) 0 0
\(65\) 1.12585 + 3.09326i 0.0173208 + 0.0475886i
\(66\) 0 0
\(67\) 53.9204 45.2446i 0.804783 0.675293i −0.144574 0.989494i \(-0.546181\pi\)
0.949356 + 0.314201i \(0.101737\pi\)
\(68\) 0 0
\(69\) −5.93319 + 84.1179i −0.0859883 + 1.21910i
\(70\) 0 0
\(71\) 38.9179 22.4692i 0.548139 0.316468i −0.200232 0.979748i \(-0.564170\pi\)
0.748371 + 0.663280i \(0.230836\pi\)
\(72\) 0 0
\(73\) 51.2495 88.7667i 0.702047 1.21598i −0.265699 0.964056i \(-0.585603\pi\)
0.967747 0.251926i \(-0.0810639\pi\)
\(74\) 0 0
\(75\) 60.6050 43.9762i 0.808067 0.586349i
\(76\) 0 0
\(77\) −63.3337 + 11.1674i −0.822516 + 0.145032i
\(78\) 0 0
\(79\) −64.7058 54.2946i −0.819060 0.687273i 0.133692 0.991023i \(-0.457317\pi\)
−0.952752 + 0.303750i \(0.901761\pi\)
\(80\) 0 0
\(81\) 45.1321 + 67.2614i 0.557187 + 0.830387i
\(82\) 0 0
\(83\) 44.2042 52.6805i 0.532580 0.634704i −0.430927 0.902387i \(-0.641813\pi\)
0.963507 + 0.267682i \(0.0862578\pi\)
\(84\) 0 0
\(85\) −0.210104 1.19156i −0.00247181 0.0140183i
\(86\) 0 0
\(87\) −73.0020 100.606i −0.839103 1.15640i
\(88\) 0 0
\(89\) −119.245 68.8461i −1.33983 0.773551i −0.353048 0.935605i \(-0.614855\pi\)
−0.986782 + 0.162054i \(0.948188\pi\)
\(90\) 0 0
\(91\) 69.8014 + 120.900i 0.767048 + 1.32857i
\(92\) 0 0
\(93\) −40.6042 2.86398i −0.436604 0.0307955i
\(94\) 0 0
\(95\) −0.0490285 0.0584299i −0.000516090 0.000615052i
\(96\) 0 0
\(97\) 112.916 41.0979i 1.16408 0.423690i 0.313526 0.949580i \(-0.398490\pi\)
0.850553 + 0.525889i \(0.176267\pi\)
\(98\) 0 0
\(99\) −9.54147 + 67.3007i −0.0963785 + 0.679805i
\(100\) 0 0
\(101\) 13.1189 + 2.31321i 0.129890 + 0.0229031i 0.238215 0.971212i \(-0.423438\pi\)
−0.108325 + 0.994116i \(0.534549\pi\)
\(102\) 0 0
\(103\) 183.161 + 66.6652i 1.77826 + 0.647235i 0.999809 + 0.0195327i \(0.00621784\pi\)
0.778454 + 0.627702i \(0.216004\pi\)
\(104\) 0 0
\(105\) −3.56510 + 3.68731i −0.0339533 + 0.0351172i
\(106\) 0 0
\(107\) 155.702i 1.45516i −0.686024 0.727579i \(-0.740645\pi\)
0.686024 0.727579i \(-0.259355\pi\)
\(108\) 0 0
\(109\) 70.1664 0.643729 0.321864 0.946786i \(-0.395691\pi\)
0.321864 + 0.946786i \(0.395691\pi\)
\(110\) 0 0
\(111\) −13.0479 + 3.73291i −0.117549 + 0.0336298i
\(112\) 0 0
\(113\) 35.8589 98.5216i 0.317336 0.871873i −0.673787 0.738925i \(-0.735334\pi\)
0.991123 0.132947i \(-0.0424441\pi\)
\(114\) 0 0
\(115\) −0.980025 + 5.55800i −0.00852196 + 0.0483304i
\(116\) 0 0
\(117\) 144.367 30.5037i 1.23390 0.260716i
\(118\) 0 0
\(119\) −17.5501 48.2184i −0.147480 0.405197i
\(120\) 0 0
\(121\) 48.9945 41.1113i 0.404913 0.339763i
\(122\) 0 0
\(123\) −207.026 + 100.819i −1.68314 + 0.819664i
\(124\) 0 0
\(125\) 8.68705 5.01547i 0.0694964 0.0401238i
\(126\) 0 0
\(127\) −77.8451 + 134.832i −0.612954 + 1.06167i 0.377786 + 0.925893i \(0.376685\pi\)
−0.990740 + 0.135774i \(0.956648\pi\)
\(128\) 0 0
\(129\) 0.544026 + 5.20633i 0.00421726 + 0.0403592i
\(130\) 0 0
\(131\) 140.642 24.7989i 1.07360 0.189305i 0.391216 0.920299i \(-0.372055\pi\)
0.682383 + 0.730994i \(0.260943\pi\)
\(132\) 0 0
\(133\) −2.47798 2.07928i −0.0186315 0.0156336i
\(134\) 0 0
\(135\) 2.53636 + 4.79114i 0.0187878 + 0.0354899i
\(136\) 0 0
\(137\) 123.444 147.114i 0.901048 1.07383i −0.0958717 0.995394i \(-0.530564\pi\)
0.996919 0.0784331i \(-0.0249917\pi\)
\(138\) 0 0
\(139\) 12.4043 + 70.3485i 0.0892398 + 0.506104i 0.996361 + 0.0852345i \(0.0271639\pi\)
−0.907121 + 0.420870i \(0.861725\pi\)
\(140\) 0 0
\(141\) −93.5245 + 209.718i −0.663295 + 1.48736i
\(142\) 0 0
\(143\) 107.235 + 61.9123i 0.749897 + 0.432953i
\(144\) 0 0
\(145\) −4.15957 7.20458i −0.0286867 0.0496868i
\(146\) 0 0
\(147\) −39.4676 + 58.4369i −0.268487 + 0.397530i
\(148\) 0 0
\(149\) −64.1576 76.4600i −0.430588 0.513155i 0.506504 0.862238i \(-0.330938\pi\)
−0.937092 + 0.349083i \(0.886493\pi\)
\(150\) 0 0
\(151\) 91.0224 33.1295i 0.602797 0.219400i −0.0225513 0.999746i \(-0.507179\pi\)
0.625349 + 0.780345i \(0.284957\pi\)
\(152\) 0 0
\(153\) −54.2048 + 1.82727i −0.354280 + 0.0119429i
\(154\) 0 0
\(155\) −2.68288 0.473063i −0.0173089 0.00305202i
\(156\) 0 0
\(157\) 225.856 + 82.2049i 1.43857 + 0.523598i 0.939376 0.342890i \(-0.111406\pi\)
0.499198 + 0.866488i \(0.333628\pi\)
\(158\) 0 0
\(159\) 249.931 + 62.4754i 1.57190 + 0.392927i
\(160\) 0 0
\(161\) 239.348i 1.48663i
\(162\) 0 0
\(163\) 58.5417 0.359152 0.179576 0.983744i \(-0.442527\pi\)
0.179576 + 0.983744i \(0.442527\pi\)
\(164\) 0 0
\(165\) −1.10324 + 4.41347i −0.00668628 + 0.0267483i
\(166\) 0 0
\(167\) −22.4027 + 61.5510i −0.134148 + 0.368569i −0.988519 0.151094i \(-0.951720\pi\)
0.854371 + 0.519663i \(0.173943\pi\)
\(168\) 0 0
\(169\) 17.3288 98.2765i 0.102537 0.581518i
\(170\) 0 0
\(171\) −2.90168 + 1.80830i −0.0169689 + 0.0105748i
\(172\) 0 0
\(173\) 25.8686 + 71.0734i 0.149530 + 0.410829i 0.991731 0.128334i \(-0.0409629\pi\)
−0.842201 + 0.539163i \(0.818741\pi\)
\(174\) 0 0
\(175\) 162.809 136.613i 0.930337 0.780646i
\(176\) 0 0
\(177\) 45.3148 + 30.6051i 0.256016 + 0.172910i
\(178\) 0 0
\(179\) −259.614 + 149.888i −1.45035 + 0.837363i −0.998501 0.0547292i \(-0.982570\pi\)
−0.451854 + 0.892092i \(0.649237\pi\)
\(180\) 0 0
\(181\) −148.560 + 257.314i −0.820774 + 1.42162i 0.0843330 + 0.996438i \(0.473124\pi\)
−0.905107 + 0.425184i \(0.860209\pi\)
\(182\) 0 0
\(183\) 102.823 + 45.8541i 0.561872 + 0.250569i
\(184\) 0 0
\(185\) −0.894495 + 0.157724i −0.00483511 + 0.000852560i
\(186\) 0 0
\(187\) −34.8654 29.2555i −0.186446 0.156447i
\(188\) 0 0
\(189\) 141.215 + 181.424i 0.747170 + 0.959917i
\(190\) 0 0
\(191\) 52.6139 62.7028i 0.275466 0.328287i −0.610519 0.792001i \(-0.709039\pi\)
0.885985 + 0.463714i \(0.153484\pi\)
\(192\) 0 0
\(193\) 20.4421 + 115.933i 0.105918 + 0.600688i 0.990850 + 0.134968i \(0.0430933\pi\)
−0.884932 + 0.465720i \(0.845796\pi\)
\(194\) 0 0
\(195\) 9.82186 1.02632i 0.0503685 0.00526317i
\(196\) 0 0
\(197\) −69.8603 40.3339i −0.354621 0.204740i 0.312098 0.950050i \(-0.398968\pi\)
−0.666719 + 0.745310i \(0.732302\pi\)
\(198\) 0 0
\(199\) −87.9561 152.345i −0.441991 0.765550i 0.555846 0.831285i \(-0.312394\pi\)
−0.997837 + 0.0657347i \(0.979061\pi\)
\(200\) 0 0
\(201\) −92.4536 189.849i −0.459968 0.944524i
\(202\) 0 0
\(203\) −226.782 270.268i −1.11715 1.33137i
\(204\) 0 0
\(205\) −14.4819 + 5.27097i −0.0706432 + 0.0257120i
\(206\) 0 0
\(207\) 240.504 + 78.4661i 1.16186 + 0.379063i
\(208\) 0 0
\(209\) −2.82559 0.498227i −0.0135196 0.00238386i
\(210\) 0 0
\(211\) 34.0107 + 12.3789i 0.161188 + 0.0586678i 0.421354 0.906896i \(-0.361555\pi\)
−0.260166 + 0.965564i \(0.583777\pi\)
\(212\) 0 0
\(213\) −37.0818 129.615i −0.174093 0.608523i
\(214\) 0 0
\(215\) 0.350341i 0.00162949i
\(216\) 0 0
\(217\) −115.535 −0.532418
\(218\) 0 0
\(219\) −221.065 213.739i −1.00943 0.975976i
\(220\) 0 0
\(221\) −33.7911 + 92.8403i −0.152901 + 0.420092i
\(222\) 0 0
\(223\) 70.0960 397.534i 0.314332 1.78266i −0.261611 0.965174i \(-0.584254\pi\)
0.575942 0.817490i \(-0.304635\pi\)
\(224\) 0 0
\(225\) −83.8987 208.382i −0.372883 0.926140i
\(226\) 0 0
\(227\) 59.9958 + 164.837i 0.264299 + 0.726155i 0.998866 + 0.0476187i \(0.0151632\pi\)
−0.734567 + 0.678536i \(0.762615\pi\)
\(228\) 0 0
\(229\) 102.807 86.2654i 0.448939 0.376705i −0.390103 0.920771i \(-0.627561\pi\)
0.839042 + 0.544066i \(0.183116\pi\)
\(230\) 0 0
\(231\) −13.5746 + 192.454i −0.0587645 + 0.833135i
\(232\) 0 0
\(233\) 342.572 197.784i 1.47027 0.848858i 0.470823 0.882228i \(-0.343957\pi\)
0.999443 + 0.0333693i \(0.0106237\pi\)
\(234\) 0 0
\(235\) −7.68412 + 13.3093i −0.0326984 + 0.0566352i
\(236\) 0 0
\(237\) −205.097 + 148.822i −0.865386 + 0.627941i
\(238\) 0 0
\(239\) −442.556 + 78.0346i −1.85170 + 0.326504i −0.985030 0.172380i \(-0.944854\pi\)
−0.866668 + 0.498885i \(0.833743\pi\)
\(240\) 0 0
\(241\) −174.607 146.513i −0.724511 0.607937i 0.204118 0.978946i \(-0.434567\pi\)
−0.928629 + 0.371009i \(0.879012\pi\)
\(242\) 0 0
\(243\) 228.595 82.4204i 0.940722 0.339179i
\(244\) 0 0
\(245\) −3.03360 + 3.61530i −0.0123820 + 0.0147563i
\(246\) 0 0
\(247\) 1.08153 + 6.13365i 0.00437866 + 0.0248326i
\(248\) 0 0
\(249\) −121.164 166.980i −0.486603 0.670603i
\(250\) 0 0
\(251\) 307.522 + 177.548i 1.22519 + 0.707362i 0.966019 0.258471i \(-0.0832186\pi\)
0.259167 + 0.965832i \(0.416552\pi\)
\(252\) 0 0
\(253\) 106.148 + 183.854i 0.419558 + 0.726697i
\(254\) 0 0
\(255\) −3.62083 0.255392i −0.0141993 0.00100154i
\(256\) 0 0
\(257\) −122.816 146.366i −0.477883 0.569519i 0.472210 0.881486i \(-0.343456\pi\)
−0.950093 + 0.311968i \(0.899012\pi\)
\(258\) 0 0
\(259\) −36.1972 + 13.1747i −0.139758 + 0.0508676i
\(260\) 0 0
\(261\) −345.920 + 139.275i −1.32537 + 0.533620i
\(262\) 0 0
\(263\) 59.8335 + 10.5503i 0.227504 + 0.0401150i 0.286238 0.958159i \(-0.407595\pi\)
−0.0587341 + 0.998274i \(0.518706\pi\)
\(264\) 0 0
\(265\) 16.2020 + 5.89705i 0.0611397 + 0.0222530i
\(266\) 0 0
\(267\) −287.127 + 296.969i −1.07538 + 1.11224i
\(268\) 0 0
\(269\) 1.02338i 0.00380438i 0.999998 + 0.00190219i \(0.000605486\pi\)
−0.999998 + 0.00190219i \(0.999395\pi\)
\(270\) 0 0
\(271\) −264.552 −0.976205 −0.488102 0.872786i \(-0.662311\pi\)
−0.488102 + 0.872786i \(0.662311\pi\)
\(272\) 0 0
\(273\) 402.654 115.196i 1.47492 0.421963i
\(274\) 0 0
\(275\) 64.4747 177.143i 0.234453 0.644155i
\(276\) 0 0
\(277\) 63.0965 357.838i 0.227785 1.29183i −0.629503 0.776998i \(-0.716742\pi\)
0.857288 0.514837i \(-0.172147\pi\)
\(278\) 0 0
\(279\) −37.8760 + 116.093i −0.135756 + 0.416103i
\(280\) 0 0
\(281\) −64.8213 178.095i −0.230681 0.633790i 0.769306 0.638880i \(-0.220602\pi\)
−0.999987 + 0.00509010i \(0.998380\pi\)
\(282\) 0 0
\(283\) −219.711 + 184.359i −0.776364 + 0.651447i −0.942330 0.334685i \(-0.891370\pi\)
0.165966 + 0.986131i \(0.446926\pi\)
\(284\) 0 0
\(285\) −0.205727 + 0.100186i −0.000721848 + 0.000351529i
\(286\) 0 0
\(287\) −566.021 + 326.792i −1.97220 + 1.13865i
\(288\) 0 0
\(289\) −126.343 + 218.832i −0.437172 + 0.757203i
\(290\) 0 0
\(291\) −37.4644 358.535i −0.128744 1.23208i
\(292\) 0 0
\(293\) 197.305 34.7903i 0.673397 0.118738i 0.173515 0.984831i \(-0.444488\pi\)
0.499883 + 0.866093i \(0.333376\pi\)
\(294\) 0 0
\(295\) 2.80348 + 2.35240i 0.00950334 + 0.00797425i
\(296\) 0 0
\(297\) 188.933 + 76.7329i 0.636139 + 0.258360i
\(298\) 0 0
\(299\) 296.225 353.027i 0.990717 1.18069i
\(300\) 0 0
\(301\) 2.58003 + 14.6321i 0.00857152 + 0.0486115i
\(302\) 0 0
\(303\) 16.2768 36.4989i 0.0537189 0.120459i
\(304\) 0 0
\(305\) 6.52540 + 3.76744i 0.0213948 + 0.0123523i
\(306\) 0 0
\(307\) 87.7799 + 152.039i 0.285928 + 0.495242i 0.972834 0.231505i \(-0.0743649\pi\)
−0.686906 + 0.726746i \(0.741032\pi\)
\(308\) 0 0
\(309\) 327.280 484.580i 1.05916 1.56822i
\(310\) 0 0
\(311\) −92.2362 109.923i −0.296579 0.353450i 0.597091 0.802174i \(-0.296323\pi\)
−0.893670 + 0.448724i \(0.851879\pi\)
\(312\) 0 0
\(313\) −469.902 + 171.030i −1.50128 + 0.546422i −0.956392 0.292085i \(-0.905651\pi\)
−0.544891 + 0.838507i \(0.683429\pi\)
\(314\) 0 0
\(315\) 8.13801 + 13.0586i 0.0258350 + 0.0414560i
\(316\) 0 0
\(317\) −159.214 28.0736i −0.502251 0.0885604i −0.0832179 0.996531i \(-0.526520\pi\)
−0.419033 + 0.907971i \(0.637631\pi\)
\(318\) 0 0
\(319\) −294.063 107.030i −0.921827 0.335518i
\(320\) 0 0
\(321\) −453.162 113.277i −1.41172 0.352888i
\(322\) 0 0
\(323\) 2.28929i 0.00708759i
\(324\) 0 0
\(325\) −409.211 −1.25911
\(326\) 0 0
\(327\) 51.0478 204.216i 0.156110 0.624513i
\(328\) 0 0
\(329\) −222.915 + 612.453i −0.677553 + 1.86156i
\(330\) 0 0
\(331\) −79.9019 + 453.146i −0.241396 + 1.36902i 0.587321 + 0.809354i \(0.300183\pi\)
−0.828717 + 0.559668i \(0.810928\pi\)
\(332\) 0 0
\(333\) 1.37172 + 40.6912i 0.00411927 + 0.122196i
\(334\) 0 0
\(335\) −4.83363 13.2803i −0.0144287 0.0396426i
\(336\) 0 0
\(337\) 262.855 220.561i 0.779984 0.654484i −0.163261 0.986583i \(-0.552201\pi\)
0.943245 + 0.332099i \(0.107757\pi\)
\(338\) 0 0
\(339\) −260.654 176.042i −0.768890 0.519299i
\(340\) 0 0
\(341\) −88.7474 + 51.2384i −0.260256 + 0.150259i
\(342\) 0 0
\(343\) 108.543 188.002i 0.316453 0.548112i
\(344\) 0 0
\(345\) 15.4633 + 6.89589i 0.0448211 + 0.0199881i
\(346\) 0 0
\(347\) 270.540 47.7035i 0.779653 0.137474i 0.230362 0.973105i \(-0.426009\pi\)
0.549291 + 0.835631i \(0.314898\pi\)
\(348\) 0 0
\(349\) −377.022 316.359i −1.08029 0.906473i −0.0843479 0.996436i \(-0.526881\pi\)
−0.995945 + 0.0899629i \(0.971325\pi\)
\(350\) 0 0
\(351\) 16.2509 442.364i 0.0462987 1.26030i
\(352\) 0 0
\(353\) 71.2407 84.9014i 0.201815 0.240514i −0.655639 0.755075i \(-0.727601\pi\)
0.857454 + 0.514561i \(0.172045\pi\)
\(354\) 0 0
\(355\) −1.56679 8.88570i −0.00441349 0.0250302i
\(356\) 0 0
\(357\) −153.105 + 15.9985i −0.428867 + 0.0448136i
\(358\) 0 0
\(359\) 480.926 + 277.663i 1.33963 + 0.773433i 0.986752 0.162237i \(-0.0518711\pi\)
0.352874 + 0.935671i \(0.385204\pi\)
\(360\) 0 0
\(361\) 180.428 + 312.510i 0.499800 + 0.865679i
\(362\) 0 0
\(363\) −84.0075 172.505i −0.231426 0.475222i
\(364\) 0 0
\(365\) −13.2284 15.7650i −0.0362423 0.0431919i
\(366\) 0 0
\(367\) −374.715 + 136.385i −1.02102 + 0.371622i −0.797656 0.603113i \(-0.793927\pi\)
−0.223367 + 0.974734i \(0.571705\pi\)
\(368\) 0 0
\(369\) 142.811 + 675.888i 0.387021 + 1.83167i
\(370\) 0 0
\(371\) 720.108 + 126.975i 1.94099 + 0.342249i
\(372\) 0 0
\(373\) 189.416 + 68.9418i 0.507818 + 0.184830i 0.583207 0.812324i \(-0.301798\pi\)
−0.0753893 + 0.997154i \(0.524020\pi\)
\(374\) 0 0
\(375\) −8.27722 28.9321i −0.0220726 0.0771523i
\(376\) 0 0
\(377\) 679.305i 1.80187i
\(378\) 0 0
\(379\) −135.501 −0.357522 −0.178761 0.983892i \(-0.557209\pi\)
−0.178761 + 0.983892i \(0.557209\pi\)
\(380\) 0 0
\(381\) 335.786 + 324.658i 0.881329 + 0.852120i
\(382\) 0 0
\(383\) −82.3154 + 226.160i −0.214923 + 0.590495i −0.999566 0.0294519i \(-0.990624\pi\)
0.784643 + 0.619947i \(0.212846\pi\)
\(384\) 0 0
\(385\) −2.24221 + 12.7162i −0.00582392 + 0.0330291i
\(386\) 0 0
\(387\) 15.5486 + 2.20438i 0.0401771 + 0.00569606i
\(388\) 0 0
\(389\) 34.2832 + 94.1925i 0.0881317 + 0.242140i 0.975927 0.218100i \(-0.0699857\pi\)
−0.887795 + 0.460240i \(0.847764\pi\)
\(390\) 0 0
\(391\) −129.760 + 108.882i −0.331867 + 0.278469i
\(392\) 0 0
\(393\) 30.1443 427.372i 0.0767031 1.08746i
\(394\) 0 0
\(395\) −14.6873 + 8.47971i −0.0371830 + 0.0214676i
\(396\) 0 0
\(397\) −31.5616 + 54.6663i −0.0795003 + 0.137699i −0.903034 0.429568i \(-0.858666\pi\)
0.823534 + 0.567267i \(0.191999\pi\)
\(398\) 0 0
\(399\) −7.85442 + 5.69932i −0.0196853 + 0.0142840i
\(400\) 0 0
\(401\) −382.225 + 67.3965i −0.953178 + 0.168071i −0.628549 0.777770i \(-0.716351\pi\)
−0.324630 + 0.945841i \(0.605240\pi\)
\(402\) 0 0
\(403\) 170.408 + 142.989i 0.422848 + 0.354812i
\(404\) 0 0
\(405\) 15.7896 3.89627i 0.0389867 0.00962042i
\(406\) 0 0
\(407\) −21.9619 + 26.1732i −0.0539604 + 0.0643075i
\(408\) 0 0
\(409\) −90.7835 514.859i −0.221965 1.25882i −0.868404 0.495858i \(-0.834854\pi\)
0.646439 0.762966i \(-0.276258\pi\)
\(410\) 0 0
\(411\) −338.360 466.305i −0.823261 1.13456i
\(412\) 0 0
\(413\) 134.412 + 77.6028i 0.325453 + 0.187900i
\(414\) 0 0
\(415\) −6.90379 11.9577i −0.0166356 0.0288138i
\(416\) 0 0
\(417\) 213.770 + 15.0781i 0.512638 + 0.0361585i
\(418\) 0 0
\(419\) −324.985 387.301i −0.775619 0.924347i 0.223107 0.974794i \(-0.428380\pi\)
−0.998727 + 0.0504466i \(0.983936\pi\)
\(420\) 0 0
\(421\) −61.4522 + 22.3668i −0.145967 + 0.0531277i −0.413971 0.910290i \(-0.635858\pi\)
0.268003 + 0.963418i \(0.413636\pi\)
\(422\) 0 0
\(423\) 542.333 + 424.774i 1.28211 + 1.00419i
\(424\) 0 0
\(425\) 148.126 + 26.1187i 0.348533 + 0.0614557i
\(426\) 0 0
\(427\) 300.279 + 109.293i 0.703230 + 0.255955i
\(428\) 0 0
\(429\) 258.209 267.060i 0.601885 0.622517i
\(430\) 0 0
\(431\) 449.973i 1.04402i 0.852939 + 0.522010i \(0.174818\pi\)
−0.852939 + 0.522010i \(0.825182\pi\)
\(432\) 0 0
\(433\) 57.6415 0.133121 0.0665606 0.997782i \(-0.478797\pi\)
0.0665606 + 0.997782i \(0.478797\pi\)
\(434\) 0 0
\(435\) −23.9948 + 6.86469i −0.0551604 + 0.0157809i
\(436\) 0 0
\(437\) −3.65221 + 10.0344i −0.00835746 + 0.0229619i
\(438\) 0 0
\(439\) 40.0913 227.369i 0.0913241 0.517925i −0.904488 0.426499i \(-0.859747\pi\)
0.995812 0.0914252i \(-0.0291422\pi\)
\(440\) 0 0
\(441\) 141.364 + 157.383i 0.320553 + 0.356877i
\(442\) 0 0
\(443\) −97.4032 267.613i −0.219872 0.604093i 0.779890 0.625917i \(-0.215275\pi\)
−0.999762 + 0.0218237i \(0.993053\pi\)
\(444\) 0 0
\(445\) −21.1780 + 17.7705i −0.0475910 + 0.0399336i
\(446\) 0 0
\(447\) −269.209 + 131.101i −0.602258 + 0.293290i
\(448\) 0 0
\(449\) 249.959 144.314i 0.556701 0.321412i −0.195119 0.980780i \(-0.562509\pi\)
0.751821 + 0.659368i \(0.229176\pi\)
\(450\) 0 0
\(451\) −289.858 + 502.048i −0.642700 + 1.11319i
\(452\) 0 0
\(453\) −30.2005 289.019i −0.0666677 0.638010i
\(454\) 0 0
\(455\) 27.6037 4.86728i 0.0606675 0.0106973i
\(456\) 0 0
\(457\) 57.0131 + 47.8396i 0.124755 + 0.104682i 0.703031 0.711160i \(-0.251830\pi\)
−0.578276 + 0.815842i \(0.696274\pi\)
\(458\) 0 0
\(459\) −34.1172 + 159.090i −0.0743294 + 0.346601i
\(460\) 0 0
\(461\) −81.2197 + 96.7939i −0.176182 + 0.209965i −0.846907 0.531740i \(-0.821538\pi\)
0.670726 + 0.741705i \(0.265983\pi\)
\(462\) 0 0
\(463\) −7.85137 44.5273i −0.0169576 0.0961713i 0.975154 0.221527i \(-0.0711040\pi\)
−0.992112 + 0.125355i \(0.959993\pi\)
\(464\) 0 0
\(465\) −3.32869 + 7.46421i −0.00715846 + 0.0160521i
\(466\) 0 0
\(467\) 395.495 + 228.339i 0.846884 + 0.488949i 0.859598 0.510970i \(-0.170714\pi\)
−0.0127140 + 0.999919i \(0.504047\pi\)
\(468\) 0 0
\(469\) −299.678 519.057i −0.638972 1.10673i
\(470\) 0 0
\(471\) 403.569 597.536i 0.856835 1.26865i
\(472\) 0 0
\(473\) 8.47099 + 10.0953i 0.0179091 + 0.0213432i
\(474\) 0 0
\(475\) 8.91013 3.24302i 0.0187582 0.00682742i
\(476\) 0 0
\(477\) 363.663 681.960i 0.762396 1.42969i
\(478\) 0 0
\(479\) 370.216 + 65.2790i 0.772893 + 0.136282i 0.546166 0.837677i \(-0.316087\pi\)
0.226726 + 0.973959i \(0.427198\pi\)
\(480\) 0 0
\(481\) 69.6945 + 25.3667i 0.144895 + 0.0527375i
\(482\) 0 0
\(483\) 696.610 + 174.132i 1.44226 + 0.360521i
\(484\) 0 0
\(485\) 24.1263i 0.0497449i
\(486\) 0 0
\(487\) −51.3988 −0.105542 −0.0527708 0.998607i \(-0.516805\pi\)
−0.0527708 + 0.998607i \(0.516805\pi\)
\(488\) 0 0
\(489\) 42.5906 170.383i 0.0870973 0.348431i
\(490\) 0 0
\(491\) −254.558 + 699.393i −0.518449 + 1.42443i 0.353780 + 0.935329i \(0.384896\pi\)
−0.872229 + 0.489098i \(0.837326\pi\)
\(492\) 0 0
\(493\) 43.3579 245.895i 0.0879471 0.498773i
\(494\) 0 0
\(495\) 12.0425 + 6.42182i 0.0243284 + 0.0129734i
\(496\) 0 0
\(497\) −130.875 359.575i −0.263329 0.723491i
\(498\) 0 0
\(499\) 292.343 245.305i 0.585858 0.491593i −0.301007 0.953622i \(-0.597323\pi\)
0.886865 + 0.462029i \(0.152878\pi\)
\(500\) 0 0
\(501\) 162.842 + 109.982i 0.325035 + 0.219525i
\(502\) 0 0
\(503\) −169.431 + 97.8211i −0.336841 + 0.194475i −0.658874 0.752253i \(-0.728967\pi\)
0.322033 + 0.946728i \(0.395634\pi\)
\(504\) 0 0
\(505\) 1.33733 2.31632i 0.00264817 0.00458677i
\(506\) 0 0
\(507\) −273.422 121.933i −0.539293 0.240499i
\(508\) 0 0
\(509\) 429.935 75.8091i 0.844666 0.148937i 0.265463 0.964121i \(-0.414475\pi\)
0.579203 + 0.815184i \(0.303364\pi\)
\(510\) 0 0
\(511\) −668.588 561.012i −1.30839 1.09787i
\(512\) 0 0
\(513\) 3.15191 + 9.76078i 0.00614408 + 0.0190269i
\(514\) 0 0
\(515\) 25.1557 29.9794i 0.0488461 0.0582125i
\(516\) 0 0
\(517\) 100.385 + 569.314i 0.194169 + 1.10119i
\(518\) 0 0
\(519\) 225.676 23.5816i 0.434828 0.0454365i
\(520\) 0 0
\(521\) 253.777 + 146.518i 0.487097 + 0.281226i 0.723369 0.690461i \(-0.242592\pi\)
−0.236272 + 0.971687i \(0.575926\pi\)
\(522\) 0 0
\(523\) −166.756 288.830i −0.318845 0.552256i 0.661402 0.750032i \(-0.269962\pi\)
−0.980247 + 0.197775i \(0.936628\pi\)
\(524\) 0 0
\(525\) −279.157 573.236i −0.531728 1.09188i
\(526\) 0 0
\(527\) −52.5577 62.6358i −0.0997300 0.118854i
\(528\) 0 0
\(529\) 245.367 89.3062i 0.463831 0.168821i
\(530\) 0 0
\(531\) 122.042 109.620i 0.229835 0.206442i
\(532\) 0 0
\(533\) 1239.30 + 218.522i 2.32514 + 0.409985i
\(534\) 0 0
\(535\) −29.3766 10.6922i −0.0549096 0.0199855i
\(536\) 0 0
\(537\) 247.366 + 864.639i 0.460644 + 1.61013i
\(538\) 0 0
\(539\) 177.528i 0.329365i
\(540\) 0 0
\(541\) −51.1409 −0.0945304 −0.0472652 0.998882i \(-0.515051\pi\)
−0.0472652 + 0.998882i \(0.515051\pi\)
\(542\) 0 0
\(543\) 640.817 + 619.578i 1.18014 + 1.14103i
\(544\) 0 0
\(545\) 4.81840 13.2385i 0.00884110 0.0242907i
\(546\) 0 0
\(547\) 78.2579 443.823i 0.143068 0.811376i −0.825831 0.563918i \(-0.809293\pi\)
0.968898 0.247459i \(-0.0795954\pi\)
\(548\) 0 0
\(549\) 208.262 265.900i 0.379348 0.484335i
\(550\) 0 0
\(551\) −5.38353 14.7911i −0.00977047 0.0268441i
\(552\) 0 0
\(553\) −550.970 + 462.319i −0.996330 + 0.836020i
\(554\) 0 0
\(555\) −0.191721 + 2.71813i −0.000345443 + 0.00489753i
\(556\) 0 0
\(557\) −255.303 + 147.399i −0.458354 + 0.264631i −0.711352 0.702836i \(-0.751917\pi\)
0.252998 + 0.967467i \(0.418583\pi\)
\(558\) 0 0
\(559\) 14.3037 24.7747i 0.0255880 0.0443196i
\(560\) 0 0
\(561\) −110.512 + 80.1897i −0.196991 + 0.142941i
\(562\) 0 0
\(563\) 570.607 100.613i 1.01351 0.178709i 0.357861 0.933775i \(-0.383506\pi\)
0.655650 + 0.755065i \(0.272395\pi\)
\(564\) 0 0
\(565\) −16.1258 13.5312i −0.0285413 0.0239490i
\(566\) 0 0
\(567\) 630.763 279.009i 1.11246 0.492079i
\(568\) 0 0
\(569\) 134.264 160.010i 0.235965 0.281212i −0.635047 0.772473i \(-0.719019\pi\)
0.871012 + 0.491261i \(0.163464\pi\)
\(570\) 0 0
\(571\) 83.0575 + 471.043i 0.145460 + 0.824943i 0.966997 + 0.254788i \(0.0820058\pi\)
−0.821537 + 0.570155i \(0.806883\pi\)
\(572\) 0 0
\(573\) −144.215 198.748i −0.251685 0.346855i
\(574\) 0 0
\(575\) −607.596 350.795i −1.05669 0.610079i
\(576\) 0 0
\(577\) 94.3702 + 163.454i 0.163553 + 0.283282i 0.936141 0.351626i \(-0.114371\pi\)
−0.772587 + 0.634908i \(0.781038\pi\)
\(578\) 0 0
\(579\) 352.288 + 24.8484i 0.608443 + 0.0429161i
\(580\) 0 0
\(581\) −376.399 448.575i −0.647847 0.772074i
\(582\) 0 0
\(583\) 609.460 221.825i 1.04539 0.380489i
\(584\) 0 0
\(585\) 4.15861 29.3327i 0.00710873 0.0501414i
\(586\) 0 0
\(587\) −563.429 99.3478i −0.959846 0.169247i −0.328290 0.944577i \(-0.606472\pi\)
−0.631556 + 0.775330i \(0.717583\pi\)
\(588\) 0 0
\(589\) −4.84364 1.76294i −0.00822350 0.00299311i
\(590\) 0 0
\(591\) −168.215 + 173.981i −0.284627 + 0.294384i
\(592\) 0 0
\(593\) 1075.40i 1.81349i 0.421682 + 0.906744i \(0.361440\pi\)
−0.421682 + 0.906744i \(0.638560\pi\)
\(594\) 0 0
\(595\) −10.3027 −0.0173154
\(596\) 0 0
\(597\) −507.381 + 145.157i −0.849885 + 0.243145i
\(598\) 0 0
\(599\) −145.620 + 400.088i −0.243106 + 0.667927i 0.756793 + 0.653655i \(0.226765\pi\)
−0.999898 + 0.0142721i \(0.995457\pi\)
\(600\) 0 0
\(601\) −128.421 + 728.311i −0.213679 + 1.21183i 0.669506 + 0.742807i \(0.266506\pi\)
−0.883185 + 0.469026i \(0.844605\pi\)
\(602\) 0 0
\(603\) −619.809 + 130.961i −1.02788 + 0.217183i
\(604\) 0 0
\(605\) −4.39205 12.0671i −0.00725959 0.0199455i
\(606\) 0 0
\(607\) 379.228 318.210i 0.624758 0.524235i −0.274537 0.961577i \(-0.588525\pi\)
0.899295 + 0.437342i \(0.144080\pi\)
\(608\) 0 0
\(609\) −951.592 + 463.411i −1.56255 + 0.760937i
\(610\) 0 0
\(611\) 1086.78 627.452i 1.77869 1.02693i
\(612\) 0 0
\(613\) −521.519 + 903.298i −0.850766 + 1.47357i 0.0297525 + 0.999557i \(0.490528\pi\)
−0.880518 + 0.474012i \(0.842805\pi\)
\(614\) 0 0
\(615\) 4.80496 + 45.9835i 0.00781294 + 0.0747699i
\(616\) 0 0
\(617\) 154.839 27.3024i 0.250955 0.0442502i −0.0467552 0.998906i \(-0.514888\pi\)
0.297711 + 0.954656i \(0.403777\pi\)
\(618\) 0 0
\(619\) −207.387 174.018i −0.335035 0.281128i 0.459713 0.888068i \(-0.347952\pi\)
−0.794748 + 0.606940i \(0.792397\pi\)
\(620\) 0 0
\(621\) 403.344 642.889i 0.649508 1.03525i
\(622\) 0 0
\(623\) −753.636 + 898.149i −1.20969 + 1.44165i
\(624\) 0 0
\(625\) 108.005 + 612.529i 0.172809 + 0.980046i
\(626\) 0 0
\(627\) −3.50575 + 7.86125i −0.00559131 + 0.0125379i
\(628\) 0 0
\(629\) −23.6090 13.6306i −0.0375341 0.0216703i
\(630\) 0 0
\(631\) 226.092 + 391.603i 0.358308 + 0.620608i 0.987678 0.156498i \(-0.0500205\pi\)
−0.629370 + 0.777106i \(0.716687\pi\)
\(632\) 0 0
\(633\) 60.7718 89.9805i 0.0960060 0.142149i
\(634\) 0 0
\(635\) 20.0933 + 23.9462i 0.0316430 + 0.0377106i
\(636\) 0 0
\(637\) 362.129 131.804i 0.568491 0.206914i
\(638\) 0 0
\(639\) −404.217 + 13.6263i −0.632577 + 0.0213245i
\(640\) 0 0
\(641\) 682.172 + 120.285i 1.06423 + 0.187653i 0.678233 0.734847i \(-0.262746\pi\)
0.385998 + 0.922500i \(0.373857\pi\)
\(642\) 0 0
\(643\) 738.955 + 268.958i 1.14923 + 0.418286i 0.845239 0.534389i \(-0.179458\pi\)
0.303992 + 0.952675i \(0.401681\pi\)
\(644\) 0 0
\(645\) 1.01965 + 0.254882i 0.00158085 + 0.000395166i
\(646\) 0 0
\(647\) 62.1427i 0.0960475i −0.998846 0.0480237i \(-0.984708\pi\)
0.998846 0.0480237i \(-0.0152923\pi\)
\(648\) 0 0
\(649\) 137.664 0.212117
\(650\) 0 0
\(651\) −84.0544 + 336.258i −0.129116 + 0.516525i
\(652\) 0 0
\(653\) 101.323 278.384i 0.155166 0.426315i −0.837614 0.546262i \(-0.816050\pi\)
0.992780 + 0.119947i \(0.0382725\pi\)
\(654\) 0 0
\(655\) 4.97914 28.2381i 0.00760174 0.0431116i
\(656\) 0 0
\(657\) −782.906 + 487.899i −1.19164 + 0.742617i
\(658\) 0 0
\(659\) −94.7068 260.205i −0.143713 0.394848i 0.846863 0.531811i \(-0.178488\pi\)
−0.990576 + 0.136963i \(0.956266\pi\)
\(660\) 0 0
\(661\) 489.455 410.702i 0.740477 0.621334i −0.192489 0.981299i \(-0.561656\pi\)
0.932966 + 0.359965i \(0.117212\pi\)
\(662\) 0 0
\(663\) 245.623 + 165.891i 0.370472 + 0.250212i
\(664\) 0 0
\(665\) −0.562467 + 0.324741i −0.000845815 + 0.000488332i
\(666\) 0 0
\(667\) −582.332 + 1008.63i −0.873062 + 1.51219i
\(668\) 0 0
\(669\) −1106.01 493.227i −1.65322 0.737260i
\(670\) 0 0
\(671\) 279.128 49.2179i 0.415989 0.0733500i
\(672\) 0 0
\(673\) 361.710 + 303.510i 0.537459 + 0.450981i 0.870668 0.491872i \(-0.163687\pi\)
−0.333209 + 0.942853i \(0.608132\pi\)
\(674\) 0 0
\(675\) −667.522 + 92.5801i −0.988922 + 0.137156i
\(676\) 0 0
\(677\) −198.994 + 237.152i −0.293935 + 0.350298i −0.892720 0.450612i \(-0.851206\pi\)
0.598785 + 0.800910i \(0.295650\pi\)
\(678\) 0 0
\(679\) −177.674 1007.64i −0.261670 1.48401i
\(680\) 0 0
\(681\) 523.399 54.6916i 0.768574 0.0803107i
\(682\) 0 0
\(683\) −654.635 377.954i −0.958470 0.553373i −0.0627681 0.998028i \(-0.519993\pi\)
−0.895702 + 0.444655i \(0.853326\pi\)
\(684\) 0 0
\(685\) −19.2794 33.3929i −0.0281451 0.0487487i
\(686\) 0 0
\(687\) −176.276 361.975i −0.256588 0.526892i
\(688\) 0 0
\(689\) −904.976 1078.51i −1.31346 1.56533i
\(690\) 0 0
\(691\) −711.871 + 259.100i −1.03020 + 0.374963i −0.801158 0.598452i \(-0.795783\pi\)
−0.229045 + 0.973416i \(0.573560\pi\)
\(692\) 0 0
\(693\) 550.252 + 179.523i 0.794014 + 0.259052i
\(694\) 0 0
\(695\) 14.1246 + 2.49055i 0.0203232 + 0.00358353i
\(696\) 0 0
\(697\) −434.654 158.201i −0.623608 0.226975i
\(698\) 0 0
\(699\) −326.410 1140.93i −0.466968 1.63223i
\(700\) 0 0
\(701\) 1368.47i 1.95217i −0.217397 0.976083i \(-0.569757\pi\)
0.217397 0.976083i \(-0.430243\pi\)
\(702\) 0 0
\(703\) −1.71855 −0.00244460
\(704\) 0 0
\(705\) 33.1456 + 32.0471i 0.0470150 + 0.0454568i
\(706\) 0 0
\(707\) 38.7956 106.590i 0.0548736 0.150764i
\(708\) 0 0
\(709\) 70.8018 401.537i 0.0998615 0.566343i −0.893287 0.449486i \(-0.851607\pi\)
0.993149 0.116857i \(-0.0372818\pi\)
\(710\) 0 0
\(711\) 283.926 + 705.195i 0.399333 + 0.991835i
\(712\) 0 0
\(713\) 130.444 + 358.391i 0.182951 + 0.502653i
\(714\) 0 0
\(715\) 19.0451 15.9807i 0.0266365 0.0223507i
\(716\) 0 0
\(717\) −94.8550 + 1344.81i −0.132294 + 1.87560i
\(718\) 0 0
\(719\) 214.968 124.112i 0.298982 0.172617i −0.343004 0.939334i \(-0.611444\pi\)
0.641985 + 0.766717i \(0.278111\pi\)
\(720\) 0 0
\(721\) 829.856 1437.35i 1.15098 1.99355i
\(722\) 0 0
\(723\) −553.449 + 401.593i −0.765490 + 0.555454i
\(724\) 0 0
\(725\) 1018.47 179.583i 1.40478 0.247701i
\(726\) 0 0
\(727\) −229.792 192.819i −0.316083 0.265225i 0.470918 0.882177i \(-0.343923\pi\)
−0.787001 + 0.616952i \(0.788367\pi\)
\(728\) 0 0
\(729\) −73.5714 725.278i −0.100921 0.994894i
\(730\) 0 0
\(731\) −6.75893 + 8.05498i −0.00924615 + 0.0110191i
\(732\) 0 0
\(733\) −132.659 752.346i −0.180981 1.02639i −0.931011 0.364990i \(-0.881072\pi\)
0.750031 0.661403i \(-0.230039\pi\)
\(734\) 0 0
\(735\) 8.31513 + 11.4594i 0.0113131 + 0.0155910i
\(736\) 0 0
\(737\) −460.393 265.808i −0.624685 0.360662i
\(738\) 0 0
\(739\) 437.913 + 758.487i 0.592575 + 1.02637i 0.993884 + 0.110427i \(0.0352219\pi\)
−0.401309 + 0.915943i \(0.631445\pi\)
\(740\) 0 0
\(741\) 18.6385 + 1.31465i 0.0251532 + 0.00177416i
\(742\) 0 0
\(743\) −57.9224 69.0292i −0.0779574 0.0929060i 0.725656 0.688058i \(-0.241536\pi\)
−0.803613 + 0.595152i \(0.797092\pi\)
\(744\) 0 0
\(745\) −18.8317 + 6.85416i −0.0252774 + 0.00920022i
\(746\) 0 0
\(747\) −574.137 + 231.160i −0.768591 + 0.309450i
\(748\) 0 0
\(749\) −1305.66 230.223i −1.74321 0.307374i
\(750\) 0 0
\(751\) 423.400 + 154.105i 0.563782 + 0.205200i 0.608159 0.793815i \(-0.291908\pi\)
−0.0443772 + 0.999015i \(0.514130\pi\)
\(752\) 0 0
\(753\) 740.473 765.856i 0.983364 1.01707i
\(754\) 0 0
\(755\) 19.4484i 0.0257595i
\(756\) 0 0
\(757\) 67.0196 0.0885332 0.0442666 0.999020i \(-0.485905\pi\)
0.0442666 + 0.999020i \(0.485905\pi\)
\(758\) 0 0
\(759\) 612.324 175.181i 0.806751 0.230804i
\(760\) 0 0
\(761\) −309.817 + 851.216i −0.407119 + 1.11855i 0.551579 + 0.834123i \(0.314026\pi\)
−0.958698 + 0.284427i \(0.908197\pi\)
\(762\) 0 0
\(763\) 103.749 588.391i 0.135975 0.771155i
\(764\) 0 0
\(765\) −3.37755 + 10.3524i −0.00441509 + 0.0135326i
\(766\) 0 0
\(767\) −102.207 280.813i −0.133256 0.366118i
\(768\) 0 0
\(769\) −706.730 + 593.017i −0.919024 + 0.771153i −0.973814 0.227345i \(-0.926995\pi\)
0.0547901 + 0.998498i \(0.482551\pi\)
\(770\) 0 0
\(771\) −515.343 + 250.964i −0.668409 + 0.325505i
\(772\) 0 0
\(773\) −1317.32 + 760.554i −1.70416 + 0.983900i −0.762721 + 0.646728i \(0.776137\pi\)
−0.941443 + 0.337172i \(0.890530\pi\)
\(774\) 0 0
\(775\) 169.331 293.290i 0.218491 0.378438i
\(776\) 0 0
\(777\) 12.0099 + 114.935i 0.0154568 + 0.147922i
\(778\) 0 0
\(779\) −28.7162 + 5.06344i −0.0368629 + 0.00649993i
\(780\) 0 0
\(781\) −259.998 218.164i −0.332904 0.279340i
\(782\) 0 0
\(783\) 153.686 + 1108.11i 0.196279 + 1.41521i
\(784\) 0 0
\(785\) 31.0196 36.9677i 0.0395154 0.0470926i
\(786\) 0 0
\(787\) 136.517 + 774.228i 0.173465 + 0.983771i 0.939900 + 0.341449i \(0.110918\pi\)
−0.766435 + 0.642322i \(0.777971\pi\)
\(788\) 0 0
\(789\) 74.2363 166.467i 0.0940891 0.210984i
\(790\) 0 0
\(791\) −773.146 446.376i −0.977429 0.564319i
\(792\) 0 0
\(793\) −307.633 532.836i −0.387936 0.671924i
\(794\) 0 0
\(795\) 28.9504 42.8649i 0.0364156 0.0539181i
\(796\) 0 0
\(797\) −442.583 527.450i −0.555311 0.661794i 0.413236 0.910624i \(-0.364398\pi\)
−0.968547 + 0.248830i \(0.919954\pi\)
\(798\) 0 0
\(799\) −433.441 + 157.759i −0.542479 + 0.197446i
\(800\) 0 0
\(801\) 655.420 + 1051.72i 0.818253 + 1.31301i
\(802\) 0 0
\(803\) −762.375 134.427i −0.949409 0.167406i
\(804\) 0 0
\(805\) 45.1584 + 16.4363i 0.0560973 + 0.0204178i
\(806\) 0 0
\(807\) 2.97849 + 0.744533i 0.00369081 + 0.000922593i
\(808\) 0 0
\(809\) 267.972i 0.331238i −0.986190 0.165619i \(-0.947038\pi\)
0.986190 0.165619i \(-0.0529623\pi\)
\(810\) 0 0
\(811\) −457.328 −0.563906 −0.281953 0.959428i \(-0.590982\pi\)
−0.281953 + 0.959428i \(0.590982\pi\)
\(812\) 0 0
\(813\) −192.468 + 769.963i −0.236738 + 0.947064i
\(814\) 0 0
\(815\) 4.02012 11.0452i 0.00493266 0.0135524i
\(816\) 0 0
\(817\) −0.115106 + 0.652799i −0.000140889 + 0.000799019i
\(818\) 0 0
\(819\) −42.3306 1255.71i −0.0516857 1.53323i
\(820\) 0 0
\(821\) 71.1226 + 195.408i 0.0866292 + 0.238012i 0.975441 0.220260i \(-0.0706907\pi\)
−0.888812 + 0.458272i \(0.848468\pi\)
\(822\) 0 0
\(823\) 239.194 200.707i 0.290636 0.243873i −0.485798 0.874071i \(-0.661471\pi\)
0.776434 + 0.630198i \(0.217026\pi\)
\(824\) 0 0
\(825\) −468.658 316.526i −0.568070 0.383668i
\(826\) 0 0
\(827\) 905.439 522.755i 1.09485 0.632111i 0.159985 0.987119i \(-0.448856\pi\)
0.934863 + 0.355009i \(0.115522\pi\)
\(828\) 0 0
\(829\) 370.655 641.994i 0.447111 0.774419i −0.551085 0.834449i \(-0.685786\pi\)
0.998197 + 0.0600295i \(0.0191195\pi\)
\(830\) 0 0
\(831\) −995.565 443.975i −1.19803 0.534266i
\(832\) 0 0
\(833\) −139.496 + 24.5969i −0.167462 + 0.0295281i
\(834\) 0 0
\(835\) 10.0745 + 8.45355i 0.0120653 + 0.0101240i
\(836\) 0 0
\(837\) 310.326 + 194.697i 0.370760 + 0.232612i
\(838\) 0 0
\(839\) 54.1406 64.5222i 0.0645299 0.0769037i −0.732814 0.680429i \(-0.761793\pi\)
0.797344 + 0.603525i \(0.206238\pi\)
\(840\) 0 0
\(841\) −152.076 862.466i −0.180828 1.02552i
\(842\) 0 0
\(843\) −565.495 + 59.0904i −0.670813 + 0.0700954i
\(844\) 0 0
\(845\) −17.3521 10.0182i −0.0205350 0.0118559i
\(846\) 0 0
\(847\) −272.301 471.639i −0.321488 0.556834i
\(848\) 0 0
\(849\) 376.723 + 773.583i 0.443726 + 0.911170i
\(850\) 0 0
\(851\) 81.7366 + 97.4099i 0.0960477 + 0.114465i
\(852\) 0 0
\(853\) −16.4245 + 5.97804i −0.0192550 + 0.00700825i −0.351630 0.936139i \(-0.614372\pi\)
0.332375 + 0.943147i \(0.392150\pi\)
\(854\) 0 0
\(855\) 0.141914 + 0.671644i 0.000165981 + 0.000785549i
\(856\) 0 0
\(857\) −749.943 132.235i −0.875079 0.154300i −0.281971 0.959423i \(-0.590988\pi\)
−0.593108 + 0.805123i \(0.702099\pi\)
\(858\) 0 0
\(859\) −889.121 323.614i −1.03507 0.376733i −0.232058 0.972702i \(-0.574546\pi\)
−0.803008 + 0.595969i \(0.796768\pi\)
\(860\) 0 0
\(861\) 539.318 + 1885.12i 0.626385 + 2.18946i
\(862\) 0 0
\(863\) 357.864i 0.414675i −0.978270 0.207337i \(-0.933520\pi\)
0.978270 0.207337i \(-0.0664798\pi\)
\(864\) 0 0
\(865\) 15.1860 0.0175561
\(866\) 0 0
\(867\) 544.981 + 526.919i 0.628583 + 0.607750i
\(868\) 0 0
\(869\) −218.192 + 599.478i −0.251084 + 0.689848i
\(870\) 0 0
\(871\) −200.391 + 1136.47i −0.230070 + 1.30479i
\(872\) 0 0
\(873\) −1070.75 151.805i −1.22652 0.173889i
\(874\) 0 0
\(875\) −29.2132 80.2625i −0.0333865 0.0917286i
\(876\) 0 0
\(877\) 833.488 699.380i 0.950386 0.797468i −0.0289766 0.999580i \(-0.509225\pi\)
0.979362 + 0.202112i \(0.0647804\pi\)
\(878\) 0 0
\(879\) 42.2894 599.558i 0.0481108 0.682091i
\(880\) 0 0
\(881\) −668.191 + 385.780i −0.758446 + 0.437889i −0.828738 0.559638i \(-0.810940\pi\)
0.0702915 + 0.997526i \(0.477607\pi\)
\(882\) 0 0
\(883\) 605.002 1047.89i 0.685166 1.18674i −0.288219 0.957565i \(-0.593063\pi\)
0.973385 0.229178i \(-0.0736037\pi\)
\(884\) 0 0
\(885\) 8.88615 6.44796i 0.0100409 0.00728584i
\(886\) 0 0
\(887\) 1334.02 235.224i 1.50397 0.265190i 0.639858 0.768493i \(-0.278993\pi\)
0.864110 + 0.503303i \(0.167882\pi\)
\(888\) 0 0
\(889\) 1015.55 + 852.146i 1.14235 + 0.958545i
\(890\) 0 0
\(891\) 360.781 494.056i 0.404917 0.554496i
\(892\) 0 0
\(893\) −18.6908 + 22.2749i −0.0209304 + 0.0249439i
\(894\) 0 0
\(895\) 10.4518 + 59.2748i 0.0116779 + 0.0662289i
\(896\) 0 0
\(897\) −811.955 1118.98i −0.905189 1.24747i
\(898\) 0 0
\(899\) −486.871 281.095i −0.541569 0.312675i
\(900\) 0 0
\(901\) 258.746 + 448.161i 0.287176 + 0.497404i
\(902\) 0 0
\(903\) 44.4629 + 3.13616i 0.0492391 + 0.00347304i
\(904\) 0 0
\(905\) 38.3461 + 45.6991i 0.0423714 + 0.0504963i
\(906\) 0 0
\(907\) 1446.41 526.450i 1.59472 0.580430i 0.616381 0.787448i \(-0.288598\pi\)
0.978337 + 0.207019i \(0.0663761\pi\)
\(908\) 0 0
\(909\) −94.3865 73.9267i −0.103835 0.0813275i
\(910\) 0 0
\(911\) −309.888 54.6416i −0.340162 0.0599798i 0.000957797 1.00000i \(-0.499695\pi\)
−0.341120 + 0.940020i \(0.610806\pi\)
\(912\) 0 0
\(913\) −488.067 177.642i −0.534575 0.194569i
\(914\) 0 0
\(915\) 15.7123 16.2509i 0.0171719 0.0177606i
\(916\) 0 0
\(917\) 1216.04i 1.32611i
\(918\) 0 0
\(919\) 956.851 1.04119 0.520594 0.853805i \(-0.325711\pi\)
0.520594 + 0.853805i \(0.325711\pi\)
\(920\) 0 0
\(921\) 506.364 144.866i 0.549798 0.157293i
\(922\) 0 0
\(923\) −251.987 + 692.329i −0.273009 + 0.750086i
\(924\) 0 0
\(925\) 19.6071 111.197i 0.0211969 0.120213i
\(926\) 0 0
\(927\) −1172.24 1305.08i −1.26455 1.40785i
\(928\) 0 0
\(929\) −373.128 1025.16i −0.401645 1.10351i −0.961472 0.274902i \(-0.911355\pi\)
0.559827 0.828610i \(-0.310868\pi\)
\(930\) 0 0
\(931\) −6.84040 + 5.73978i −0.00734737 + 0.00616518i
\(932\) 0 0
\(933\) −387.029 + 188.477i −0.414822 + 0.202012i
\(934\) 0 0
\(935\) −7.91394 + 4.56912i −0.00846411 + 0.00488676i
\(936\) 0 0
\(937\) −223.525 + 387.156i −0.238554 + 0.413187i −0.960299 0.278971i \(-0.910007\pi\)
0.721746 + 0.692158i \(0.243340\pi\)
\(938\) 0 0
\(939\) 155.909 + 1492.05i 0.166038 + 1.58898i
\(940\) 0 0
\(941\) −1.45674 + 0.256863i −0.00154808 + 0.000272968i −0.174422 0.984671i \(-0.555806\pi\)
0.172874 + 0.984944i \(0.444695\pi\)
\(942\) 0 0
\(943\) 1652.78 + 1386.85i 1.75268 + 1.47068i
\(944\) 0 0
\(945\) 43.9271 14.1848i 0.0464837 0.0150103i
\(946\) 0 0
\(947\) 997.650 1188.95i 1.05348 1.25549i 0.0877005 0.996147i \(-0.472048\pi\)
0.965784 0.259347i \(-0.0835074\pi\)
\(948\) 0 0
\(949\) 291.809 + 1654.93i 0.307491 + 1.74387i
\(950\) 0 0
\(951\) −197.539 + 442.958i −0.207717 + 0.465782i
\(952\) 0 0
\(953\) 1466.92 + 846.929i 1.53927 + 0.888698i 0.998882 + 0.0472820i \(0.0150559\pi\)
0.540388 + 0.841416i \(0.318277\pi\)
\(954\) 0 0
\(955\) −8.21723 14.2327i −0.00860443 0.0149033i
\(956\) 0 0
\(957\) −525.444 + 777.987i −0.549053 + 0.812944i
\(958\) 0 0
\(959\) −1051.12 1252.68i −1.09606 1.30624i
\(960\) 0 0
\(961\) 730.047 265.715i 0.759674 0.276499i
\(962\) 0 0
\(963\) −659.374 + 1236.49i −0.684708 + 1.28400i
\(964\) 0 0
\(965\) 23.2771 + 4.10438i 0.0241213 + 0.00425324i
\(966\) 0 0
\(967\) −557.509 202.917i −0.576535 0.209841i 0.0372622 0.999306i \(-0.488136\pi\)
−0.613797 + 0.789464i \(0.710359\pi\)
\(968\) 0 0
\(969\) −6.66286 1.66552i −0.00687602 0.00171880i
\(970\) 0 0
\(971\) 597.827i 0.615682i 0.951438 + 0.307841i \(0.0996065\pi\)
−0.951438 + 0.307841i \(0.900394\pi\)
\(972\) 0 0
\(973\) 608.259 0.625138
\(974\) 0 0
\(975\) −297.712 + 1190.99i −0.305345 + 1.22153i
\(976\) 0 0
\(977\) −44.3601 + 121.878i −0.0454044 + 0.124748i −0.960322 0.278892i \(-0.910033\pi\)
0.914918 + 0.403640i \(0.132255\pi\)
\(978\) 0 0
\(979\) −180.583 + 1024.14i −0.184457 + 1.04611i
\(980\) 0 0
\(981\) −557.221 297.144i −0.568013 0.302899i
\(982\) 0 0
\(983\) 145.726 + 400.380i 0.148246 + 0.407304i 0.991483 0.130240i \(-0.0415747\pi\)
−0.843236 + 0.537543i \(0.819352\pi\)
\(984\) 0 0
\(985\) −12.4073 + 10.4109i −0.0125962 + 0.0105695i
\(986\) 0 0
\(987\) 1620.34 + 1094.36i 1.64168 + 1.10877i
\(988\) 0 0
\(989\) 42.4761 24.5236i 0.0429485 0.0247963i
\(990\) 0 0
\(991\) 194.763 337.340i 0.196532 0.340403i −0.750870 0.660450i \(-0.770365\pi\)
0.947402 + 0.320047i \(0.103699\pi\)
\(992\) 0 0
\(993\) 1260.73 + 562.226i 1.26962 + 0.566189i
\(994\) 0 0
\(995\) −34.7832 + 6.13322i −0.0349580 + 0.00616404i
\(996\) 0 0
\(997\) −505.149 423.871i −0.506669 0.425146i 0.353286 0.935515i \(-0.385064\pi\)
−0.859955 + 0.510369i \(0.829509\pi\)
\(998\) 0 0
\(999\) 119.427 + 25.6115i 0.119547 + 0.0256372i
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 432.3.bc.b.65.4 36
4.3 odd 2 108.3.k.a.65.3 yes 36
12.11 even 2 324.3.k.a.197.4 36
27.5 odd 18 inner 432.3.bc.b.113.4 36
108.7 odd 18 2916.3.c.b.1457.19 36
108.47 even 18 2916.3.c.b.1457.18 36
108.59 even 18 108.3.k.a.5.3 36
108.103 odd 18 324.3.k.a.125.4 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.3.k.a.5.3 36 108.59 even 18
108.3.k.a.65.3 yes 36 4.3 odd 2
324.3.k.a.125.4 36 108.103 odd 18
324.3.k.a.197.4 36 12.11 even 2
432.3.bc.b.65.4 36 1.1 even 1 trivial
432.3.bc.b.113.4 36 27.5 odd 18 inner
2916.3.c.b.1457.18 36 108.47 even 18
2916.3.c.b.1457.19 36 108.7 odd 18