Properties

Label 768.2.o.a.95.13
Level $768$
Weight $2$
Character 768.95
Analytic conductor $6.133$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,2,Mod(95,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.o (of order \(8\), degree \(4\), 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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 95.13
Character \(\chi\) \(=\) 768.95
Dual form 768.2.o.a.671.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.47792 + 0.903198i) q^{3} +(2.81491 - 1.16597i) q^{5} +(0.543879 + 0.543879i) q^{7} +(1.36847 + 2.66970i) q^{9} +O(q^{10})\) \(q+(1.47792 + 0.903198i) q^{3} +(2.81491 - 1.16597i) q^{5} +(0.543879 + 0.543879i) q^{7} +(1.36847 + 2.66970i) q^{9} +(-3.96678 + 1.64309i) q^{11} +(-1.13904 + 2.74988i) q^{13} +(5.21329 + 0.819208i) q^{15} +5.73443 q^{17} +(0.0793211 + 0.0328559i) q^{19} +(0.312577 + 1.29504i) q^{21} +(1.46457 + 1.46457i) q^{23} +(3.02867 - 3.02867i) q^{25} +(-0.388788 + 5.18159i) q^{27} +(-0.520422 + 1.25641i) q^{29} -5.64072i q^{31} +(-7.34659 - 1.15443i) q^{33} +(2.16511 + 0.896820i) q^{35} +(-4.19628 - 10.1307i) q^{37} +(-4.16709 + 3.03532i) q^{39} +(4.93573 - 4.93573i) q^{41} +(-3.50661 - 8.46571i) q^{43} +(6.96490 + 5.91936i) q^{45} +8.98904i q^{47} -6.40839i q^{49} +(8.47500 + 5.17932i) q^{51} +(-1.04873 - 2.53186i) q^{53} +(-9.25030 + 9.25030i) q^{55} +(0.0875545 + 0.120201i) q^{57} +(-0.498592 - 1.20371i) q^{59} +(3.69353 + 1.52991i) q^{61} +(-0.707713 + 2.19627i) q^{63} +9.06875i q^{65} +(-3.35550 + 8.10090i) q^{67} +(0.841717 + 3.48732i) q^{69} +(4.08070 - 4.08070i) q^{71} +(1.59075 + 1.59075i) q^{73} +(7.21160 - 1.74063i) q^{75} +(-3.05109 - 1.26380i) q^{77} -0.637492 q^{79} +(-5.25459 + 7.30679i) q^{81} +(-1.94530 + 4.69638i) q^{83} +(16.1419 - 6.68618i) q^{85} +(-1.90393 + 1.38682i) q^{87} +(0.902588 + 0.902588i) q^{89} +(-2.11510 + 0.876104i) q^{91} +(5.09469 - 8.33651i) q^{93} +0.261590 q^{95} -13.3054 q^{97} +(-9.81497 - 8.34158i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 4 q^{3} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q - 4 q^{3} + 8 q^{7} - 4 q^{9} + 8 q^{13} + 8 q^{15} - 8 q^{19} + 4 q^{21} - 8 q^{25} - 28 q^{27} - 8 q^{33} + 8 q^{37} + 28 q^{39} - 8 q^{43} + 4 q^{45} - 16 q^{51} - 24 q^{55} - 4 q^{57} + 40 q^{61} + 56 q^{67} + 4 q^{69} - 8 q^{73} + 16 q^{75} - 16 q^{79} + 48 q^{85} - 52 q^{87} + 40 q^{91} - 8 q^{93} - 16 q^{97} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{3}{8}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.47792 + 0.903198i 0.853275 + 0.521461i
\(4\) 0 0
\(5\) 2.81491 1.16597i 1.25886 0.521438i 0.349303 0.937010i \(-0.386418\pi\)
0.909561 + 0.415571i \(0.136418\pi\)
\(6\) 0 0
\(7\) 0.543879 + 0.543879i 0.205567 + 0.205567i 0.802380 0.596813i \(-0.203567\pi\)
−0.596813 + 0.802380i \(0.703567\pi\)
\(8\) 0 0
\(9\) 1.36847 + 2.66970i 0.456156 + 0.889900i
\(10\) 0 0
\(11\) −3.96678 + 1.64309i −1.19603 + 0.495411i −0.889713 0.456520i \(-0.849096\pi\)
−0.306314 + 0.951930i \(0.599096\pi\)
\(12\) 0 0
\(13\) −1.13904 + 2.74988i −0.315912 + 0.762680i 0.683550 + 0.729903i \(0.260435\pi\)
−0.999463 + 0.0327767i \(0.989565\pi\)
\(14\) 0 0
\(15\) 5.21329 + 0.819208i 1.34607 + 0.211519i
\(16\) 0 0
\(17\) 5.73443 1.39080 0.695402 0.718621i \(-0.255226\pi\)
0.695402 + 0.718621i \(0.255226\pi\)
\(18\) 0 0
\(19\) 0.0793211 + 0.0328559i 0.0181975 + 0.00753765i 0.391764 0.920066i \(-0.371865\pi\)
−0.373566 + 0.927604i \(0.621865\pi\)
\(20\) 0 0
\(21\) 0.312577 + 1.29504i 0.0682098 + 0.282600i
\(22\) 0 0
\(23\) 1.46457 + 1.46457i 0.305385 + 0.305385i 0.843116 0.537731i \(-0.180719\pi\)
−0.537731 + 0.843116i \(0.680719\pi\)
\(24\) 0 0
\(25\) 3.02867 3.02867i 0.605733 0.605733i
\(26\) 0 0
\(27\) −0.388788 + 5.18159i −0.0748222 + 0.997197i
\(28\) 0 0
\(29\) −0.520422 + 1.25641i −0.0966400 + 0.233310i −0.964805 0.262966i \(-0.915299\pi\)
0.868165 + 0.496275i \(0.165299\pi\)
\(30\) 0 0
\(31\) 5.64072i 1.01310i −0.862210 0.506552i \(-0.830920\pi\)
0.862210 0.506552i \(-0.169080\pi\)
\(32\) 0 0
\(33\) −7.34659 1.15443i −1.27888 0.200961i
\(34\) 0 0
\(35\) 2.16511 + 0.896820i 0.365971 + 0.151590i
\(36\) 0 0
\(37\) −4.19628 10.1307i −0.689864 1.66548i −0.745055 0.667003i \(-0.767577\pi\)
0.0551907 0.998476i \(-0.482423\pi\)
\(38\) 0 0
\(39\) −4.16709 + 3.03532i −0.667269 + 0.486040i
\(40\) 0 0
\(41\) 4.93573 4.93573i 0.770831 0.770831i −0.207421 0.978252i \(-0.566507\pi\)
0.978252 + 0.207421i \(0.0665069\pi\)
\(42\) 0 0
\(43\) −3.50661 8.46571i −0.534753 1.29101i −0.928344 0.371722i \(-0.878767\pi\)
0.393591 0.919286i \(-0.371233\pi\)
\(44\) 0 0
\(45\) 6.96490 + 5.91936i 1.03827 + 0.882406i
\(46\) 0 0
\(47\) 8.98904i 1.31119i 0.755115 + 0.655593i \(0.227581\pi\)
−0.755115 + 0.655593i \(0.772419\pi\)
\(48\) 0 0
\(49\) 6.40839i 0.915485i
\(50\) 0 0
\(51\) 8.47500 + 5.17932i 1.18674 + 0.725250i
\(52\) 0 0
\(53\) −1.04873 2.53186i −0.144054 0.347777i 0.835341 0.549733i \(-0.185270\pi\)
−0.979395 + 0.201955i \(0.935270\pi\)
\(54\) 0 0
\(55\) −9.25030 + 9.25030i −1.24731 + 1.24731i
\(56\) 0 0
\(57\) 0.0875545 + 0.120201i 0.0115969 + 0.0159210i
\(58\) 0 0
\(59\) −0.498592 1.20371i −0.0649111 0.156709i 0.888095 0.459659i \(-0.152028\pi\)
−0.953006 + 0.302950i \(0.902028\pi\)
\(60\) 0 0
\(61\) 3.69353 + 1.52991i 0.472908 + 0.195885i 0.606392 0.795166i \(-0.292616\pi\)
−0.133484 + 0.991051i \(0.542616\pi\)
\(62\) 0 0
\(63\) −0.707713 + 2.19627i −0.0891634 + 0.276704i
\(64\) 0 0
\(65\) 9.06875i 1.12484i
\(66\) 0 0
\(67\) −3.35550 + 8.10090i −0.409940 + 0.989682i 0.575213 + 0.818004i \(0.304919\pi\)
−0.985153 + 0.171679i \(0.945081\pi\)
\(68\) 0 0
\(69\) 0.841717 + 3.48732i 0.101331 + 0.419824i
\(70\) 0 0
\(71\) 4.08070 4.08070i 0.484290 0.484290i −0.422209 0.906499i \(-0.638745\pi\)
0.906499 + 0.422209i \(0.138745\pi\)
\(72\) 0 0
\(73\) 1.59075 + 1.59075i 0.186183 + 0.186183i 0.794044 0.607861i \(-0.207972\pi\)
−0.607861 + 0.794044i \(0.707972\pi\)
\(74\) 0 0
\(75\) 7.21160 1.74063i 0.832724 0.200990i
\(76\) 0 0
\(77\) −3.05109 1.26380i −0.347704 0.144024i
\(78\) 0 0
\(79\) −0.637492 −0.0717235 −0.0358618 0.999357i \(-0.511418\pi\)
−0.0358618 + 0.999357i \(0.511418\pi\)
\(80\) 0 0
\(81\) −5.25459 + 7.30679i −0.583844 + 0.811866i
\(82\) 0 0
\(83\) −1.94530 + 4.69638i −0.213525 + 0.515495i −0.993960 0.109742i \(-0.964998\pi\)
0.780435 + 0.625237i \(0.214998\pi\)
\(84\) 0 0
\(85\) 16.1419 6.68618i 1.75083 0.725218i
\(86\) 0 0
\(87\) −1.90393 + 1.38682i −0.204122 + 0.148683i
\(88\) 0 0
\(89\) 0.902588 + 0.902588i 0.0956742 + 0.0956742i 0.753324 0.657650i \(-0.228449\pi\)
−0.657650 + 0.753324i \(0.728449\pi\)
\(90\) 0 0
\(91\) −2.11510 + 0.876104i −0.221723 + 0.0918406i
\(92\) 0 0
\(93\) 5.09469 8.33651i 0.528295 0.864456i
\(94\) 0 0
\(95\) 0.261590 0.0268386
\(96\) 0 0
\(97\) −13.3054 −1.35095 −0.675477 0.737381i \(-0.736062\pi\)
−0.675477 + 0.737381i \(0.736062\pi\)
\(98\) 0 0
\(99\) −9.81497 8.34158i −0.986441 0.838360i
\(100\) 0 0
\(101\) −5.49437 + 2.27584i −0.546710 + 0.226455i −0.638904 0.769286i \(-0.720612\pi\)
0.0921939 + 0.995741i \(0.470612\pi\)
\(102\) 0 0
\(103\) −7.86860 7.86860i −0.775316 0.775316i 0.203714 0.979030i \(-0.434699\pi\)
−0.979030 + 0.203714i \(0.934699\pi\)
\(104\) 0 0
\(105\) 2.38985 + 3.28095i 0.233226 + 0.320188i
\(106\) 0 0
\(107\) −0.901495 + 0.373412i −0.0871508 + 0.0360991i −0.425833 0.904802i \(-0.640019\pi\)
0.338682 + 0.940901i \(0.390019\pi\)
\(108\) 0 0
\(109\) −0.457650 + 1.10486i −0.0438349 + 0.105827i −0.944280 0.329142i \(-0.893241\pi\)
0.900446 + 0.434969i \(0.143241\pi\)
\(110\) 0 0
\(111\) 2.94829 18.7624i 0.279840 1.78085i
\(112\) 0 0
\(113\) −8.48446 −0.798151 −0.399076 0.916918i \(-0.630669\pi\)
−0.399076 + 0.916918i \(0.630669\pi\)
\(114\) 0 0
\(115\) 5.83029 + 2.41499i 0.543678 + 0.225199i
\(116\) 0 0
\(117\) −8.90010 + 0.722234i −0.822814 + 0.0667706i
\(118\) 0 0
\(119\) 3.11883 + 3.11883i 0.285903 + 0.285903i
\(120\) 0 0
\(121\) 5.25738 5.25738i 0.477944 0.477944i
\(122\) 0 0
\(123\) 11.7525 2.83665i 1.05969 0.255772i
\(124\) 0 0
\(125\) −0.835790 + 2.01777i −0.0747553 + 0.180475i
\(126\) 0 0
\(127\) 7.01144i 0.622165i −0.950383 0.311083i \(-0.899308\pi\)
0.950383 0.311083i \(-0.100692\pi\)
\(128\) 0 0
\(129\) 2.46373 15.6788i 0.216920 1.38044i
\(130\) 0 0
\(131\) 4.09398 + 1.69578i 0.357693 + 0.148161i 0.554291 0.832323i \(-0.312989\pi\)
−0.196598 + 0.980484i \(0.562989\pi\)
\(132\) 0 0
\(133\) 0.0252714 + 0.0610107i 0.00219131 + 0.00529029i
\(134\) 0 0
\(135\) 4.94718 + 15.0390i 0.425786 + 1.29435i
\(136\) 0 0
\(137\) −6.04464 + 6.04464i −0.516428 + 0.516428i −0.916489 0.400061i \(-0.868989\pi\)
0.400061 + 0.916489i \(0.368989\pi\)
\(138\) 0 0
\(139\) −3.82265 9.22869i −0.324233 0.782767i −0.998999 0.0447354i \(-0.985756\pi\)
0.674766 0.738032i \(-0.264244\pi\)
\(140\) 0 0
\(141\) −8.11888 + 13.2850i −0.683733 + 1.11880i
\(142\) 0 0
\(143\) 12.7797i 1.06869i
\(144\) 0 0
\(145\) 4.14347i 0.344097i
\(146\) 0 0
\(147\) 5.78805 9.47106i 0.477390 0.781160i
\(148\) 0 0
\(149\) 4.03660 + 9.74522i 0.330691 + 0.798359i 0.998538 + 0.0540601i \(0.0172163\pi\)
−0.667846 + 0.744299i \(0.732784\pi\)
\(150\) 0 0
\(151\) 7.98901 7.98901i 0.650136 0.650136i −0.302889 0.953026i \(-0.597951\pi\)
0.953026 + 0.302889i \(0.0979512\pi\)
\(152\) 0 0
\(153\) 7.84738 + 15.3092i 0.634423 + 1.23768i
\(154\) 0 0
\(155\) −6.57692 15.8781i −0.528271 1.27536i
\(156\) 0 0
\(157\) −2.65449 1.09953i −0.211852 0.0877518i 0.274234 0.961663i \(-0.411576\pi\)
−0.486086 + 0.873911i \(0.661576\pi\)
\(158\) 0 0
\(159\) 0.736834 4.68908i 0.0584347 0.371868i
\(160\) 0 0
\(161\) 1.59310i 0.125554i
\(162\) 0 0
\(163\) 3.51824 8.49377i 0.275570 0.665284i −0.724133 0.689660i \(-0.757760\pi\)
0.999703 + 0.0243763i \(0.00775998\pi\)
\(164\) 0 0
\(165\) −22.0260 + 5.31631i −1.71472 + 0.413874i
\(166\) 0 0
\(167\) 6.70414 6.70414i 0.518782 0.518782i −0.398421 0.917203i \(-0.630442\pi\)
0.917203 + 0.398421i \(0.130442\pi\)
\(168\) 0 0
\(169\) 2.92794 + 2.92794i 0.225226 + 0.225226i
\(170\) 0 0
\(171\) 0.0208330 + 0.256726i 0.00159314 + 0.0196323i
\(172\) 0 0
\(173\) −18.7129 7.75114i −1.42272 0.589308i −0.467174 0.884165i \(-0.654728\pi\)
−0.955541 + 0.294857i \(0.904728\pi\)
\(174\) 0 0
\(175\) 3.29446 0.249037
\(176\) 0 0
\(177\) 0.350309 2.22930i 0.0263308 0.167565i
\(178\) 0 0
\(179\) 7.00110 16.9021i 0.523286 1.26333i −0.412564 0.910928i \(-0.635367\pi\)
0.935851 0.352397i \(-0.114633\pi\)
\(180\) 0 0
\(181\) −14.4907 + 6.00226i −1.07709 + 0.446144i −0.849487 0.527610i \(-0.823088\pi\)
−0.227601 + 0.973754i \(0.573088\pi\)
\(182\) 0 0
\(183\) 4.07691 + 5.59706i 0.301374 + 0.413747i
\(184\) 0 0
\(185\) −23.6243 23.6243i −1.73689 1.73689i
\(186\) 0 0
\(187\) −22.7472 + 9.42220i −1.66344 + 0.689019i
\(188\) 0 0
\(189\) −3.02961 + 2.60670i −0.220372 + 0.189610i
\(190\) 0 0
\(191\) −15.7313 −1.13828 −0.569139 0.822241i \(-0.692723\pi\)
−0.569139 + 0.822241i \(0.692723\pi\)
\(192\) 0 0
\(193\) 12.2248 0.879962 0.439981 0.898007i \(-0.354985\pi\)
0.439981 + 0.898007i \(0.354985\pi\)
\(194\) 0 0
\(195\) −8.19087 + 13.4028i −0.586560 + 0.959797i
\(196\) 0 0
\(197\) 21.5949 8.94488i 1.53857 0.637297i 0.557365 0.830267i \(-0.311812\pi\)
0.981205 + 0.192971i \(0.0618122\pi\)
\(198\) 0 0
\(199\) 1.57778 + 1.57778i 0.111846 + 0.111846i 0.760815 0.648969i \(-0.224799\pi\)
−0.648969 + 0.760815i \(0.724799\pi\)
\(200\) 0 0
\(201\) −12.2759 + 8.94176i −0.865873 + 0.630703i
\(202\) 0 0
\(203\) −0.966382 + 0.400288i −0.0678267 + 0.0280947i
\(204\) 0 0
\(205\) 8.13869 19.6485i 0.568431 1.37231i
\(206\) 0 0
\(207\) −1.90575 + 5.91420i −0.132459 + 0.411065i
\(208\) 0 0
\(209\) −0.368634 −0.0254990
\(210\) 0 0
\(211\) −25.3543 10.5021i −1.74546 0.722994i −0.998296 0.0583569i \(-0.981414\pi\)
−0.747166 0.664637i \(-0.768586\pi\)
\(212\) 0 0
\(213\) 9.71660 2.34525i 0.665771 0.160694i
\(214\) 0 0
\(215\) −19.7415 19.7415i −1.34636 1.34636i
\(216\) 0 0
\(217\) 3.06787 3.06787i 0.208261 0.208261i
\(218\) 0 0
\(219\) 0.914229 + 3.78774i 0.0617779 + 0.255952i
\(220\) 0 0
\(221\) −6.53174 + 15.7690i −0.439372 + 1.06074i
\(222\) 0 0
\(223\) 23.6266i 1.58216i 0.611716 + 0.791078i \(0.290480\pi\)
−0.611716 + 0.791078i \(0.709520\pi\)
\(224\) 0 0
\(225\) 12.2303 + 3.94100i 0.815351 + 0.262733i
\(226\) 0 0
\(227\) 16.7455 + 6.93620i 1.11144 + 0.460372i 0.861432 0.507873i \(-0.169568\pi\)
0.250004 + 0.968245i \(0.419568\pi\)
\(228\) 0 0
\(229\) 5.05650 + 12.2075i 0.334143 + 0.806693i 0.998254 + 0.0590592i \(0.0188101\pi\)
−0.664111 + 0.747634i \(0.731190\pi\)
\(230\) 0 0
\(231\) −3.36779 4.62353i −0.221584 0.304206i
\(232\) 0 0
\(233\) 7.93372 7.93372i 0.519756 0.519756i −0.397742 0.917497i \(-0.630206\pi\)
0.917497 + 0.397742i \(0.130206\pi\)
\(234\) 0 0
\(235\) 10.4810 + 25.3033i 0.683703 + 1.65060i
\(236\) 0 0
\(237\) −0.942160 0.575782i −0.0611999 0.0374010i
\(238\) 0 0
\(239\) 16.7260i 1.08191i 0.841051 + 0.540956i \(0.181937\pi\)
−0.841051 + 0.540956i \(0.818063\pi\)
\(240\) 0 0
\(241\) 9.00218i 0.579881i −0.957045 0.289941i \(-0.906364\pi\)
0.957045 0.289941i \(-0.0936356\pi\)
\(242\) 0 0
\(243\) −14.3653 + 6.05289i −0.921536 + 0.388293i
\(244\) 0 0
\(245\) −7.47200 18.0390i −0.477369 1.15247i
\(246\) 0 0
\(247\) −0.180700 + 0.180700i −0.0114976 + 0.0114976i
\(248\) 0 0
\(249\) −7.11675 + 5.18386i −0.451006 + 0.328514i
\(250\) 0 0
\(251\) −0.601450 1.45203i −0.0379632 0.0916513i 0.903761 0.428038i \(-0.140795\pi\)
−0.941724 + 0.336387i \(0.890795\pi\)
\(252\) 0 0
\(253\) −8.21607 3.40321i −0.516540 0.213958i
\(254\) 0 0
\(255\) 29.8953 + 4.69769i 1.87211 + 0.294181i
\(256\) 0 0
\(257\) 8.28941i 0.517079i 0.966001 + 0.258539i \(0.0832412\pi\)
−0.966001 + 0.258539i \(0.916759\pi\)
\(258\) 0 0
\(259\) 3.22761 7.79215i 0.200554 0.484181i
\(260\) 0 0
\(261\) −4.06642 + 0.329986i −0.251705 + 0.0204256i
\(262\) 0 0
\(263\) 8.76126 8.76126i 0.540243 0.540243i −0.383357 0.923600i \(-0.625232\pi\)
0.923600 + 0.383357i \(0.125232\pi\)
\(264\) 0 0
\(265\) −5.90415 5.90415i −0.362689 0.362689i
\(266\) 0 0
\(267\) 0.518733 + 2.14916i 0.0317460 + 0.131527i
\(268\) 0 0
\(269\) 15.1023 + 6.25558i 0.920804 + 0.381409i 0.792182 0.610284i \(-0.208945\pi\)
0.128621 + 0.991694i \(0.458945\pi\)
\(270\) 0 0
\(271\) −17.5035 −1.06326 −0.531631 0.846976i \(-0.678421\pi\)
−0.531631 + 0.846976i \(0.678421\pi\)
\(272\) 0 0
\(273\) −3.91724 0.615548i −0.237082 0.0372547i
\(274\) 0 0
\(275\) −7.03766 + 16.9904i −0.424387 + 1.02456i
\(276\) 0 0
\(277\) −21.3684 + 8.85109i −1.28390 + 0.531811i −0.917163 0.398512i \(-0.869527\pi\)
−0.366742 + 0.930323i \(0.619527\pi\)
\(278\) 0 0
\(279\) 15.0590 7.71915i 0.901561 0.462133i
\(280\) 0 0
\(281\) 5.93438 + 5.93438i 0.354015 + 0.354015i 0.861601 0.507586i \(-0.169462\pi\)
−0.507586 + 0.861601i \(0.669462\pi\)
\(282\) 0 0
\(283\) 23.7525 9.83861i 1.41194 0.584845i 0.459119 0.888375i \(-0.348165\pi\)
0.952822 + 0.303530i \(0.0981653\pi\)
\(284\) 0 0
\(285\) 0.386608 + 0.236268i 0.0229007 + 0.0139953i
\(286\) 0 0
\(287\) 5.36888 0.316915
\(288\) 0 0
\(289\) 15.8837 0.934334
\(290\) 0 0
\(291\) −19.6642 12.0174i −1.15274 0.704471i
\(292\) 0 0
\(293\) 4.20032 1.73983i 0.245386 0.101642i −0.256601 0.966517i \(-0.582603\pi\)
0.501987 + 0.864875i \(0.332603\pi\)
\(294\) 0 0
\(295\) −2.80698 2.80698i −0.163429 0.163429i
\(296\) 0 0
\(297\) −6.97159 21.1930i −0.404533 1.22974i
\(298\) 0 0
\(299\) −5.69562 + 2.35920i −0.329386 + 0.136436i
\(300\) 0 0
\(301\) 2.69715 6.51149i 0.155461 0.375316i
\(302\) 0 0
\(303\) −10.1758 1.59900i −0.584582 0.0918602i
\(304\) 0 0
\(305\) 12.1808 0.697469
\(306\) 0 0
\(307\) 12.3645 + 5.12154i 0.705678 + 0.292302i 0.706515 0.707698i \(-0.250266\pi\)
−0.000836540 1.00000i \(0.500266\pi\)
\(308\) 0 0
\(309\) −4.52222 18.7360i −0.257260 1.06586i
\(310\) 0 0
\(311\) −10.7303 10.7303i −0.608462 0.608462i 0.334082 0.942544i \(-0.391574\pi\)
−0.942544 + 0.334082i \(0.891574\pi\)
\(312\) 0 0
\(313\) −17.8971 + 17.8971i −1.01160 + 1.01160i −0.0116708 + 0.999932i \(0.503715\pi\)
−0.999932 + 0.0116708i \(0.996285\pi\)
\(314\) 0 0
\(315\) 0.568650 + 7.00747i 0.0320398 + 0.394826i
\(316\) 0 0
\(317\) 2.59675 6.26910i 0.145848 0.352108i −0.834026 0.551725i \(-0.813970\pi\)
0.979874 + 0.199617i \(0.0639698\pi\)
\(318\) 0 0
\(319\) 5.83900i 0.326921i
\(320\) 0 0
\(321\) −1.66960 0.262358i −0.0931879 0.0146434i
\(322\) 0 0
\(323\) 0.454861 + 0.188410i 0.0253091 + 0.0104834i
\(324\) 0 0
\(325\) 4.87871 + 11.7782i 0.270622 + 0.653340i
\(326\) 0 0
\(327\) −1.67428 + 1.21955i −0.0925878 + 0.0674411i
\(328\) 0 0
\(329\) −4.88895 + 4.88895i −0.269536 + 0.269536i
\(330\) 0 0
\(331\) 4.00950 + 9.67979i 0.220382 + 0.532050i 0.994942 0.100451i \(-0.0320286\pi\)
−0.774560 + 0.632501i \(0.782029\pi\)
\(332\) 0 0
\(333\) 21.3035 25.0664i 1.16742 1.37363i
\(334\) 0 0
\(335\) 26.7157i 1.45963i
\(336\) 0 0
\(337\) 4.80498i 0.261744i 0.991399 + 0.130872i \(0.0417777\pi\)
−0.991399 + 0.130872i \(0.958222\pi\)
\(338\) 0 0
\(339\) −12.5393 7.66315i −0.681042 0.416205i
\(340\) 0 0
\(341\) 9.26823 + 22.3755i 0.501903 + 1.21170i
\(342\) 0 0
\(343\) 7.29254 7.29254i 0.393760 0.393760i
\(344\) 0 0
\(345\) 6.43547 + 8.83505i 0.346474 + 0.475663i
\(346\) 0 0
\(347\) 5.87587 + 14.1856i 0.315433 + 0.761523i 0.999485 + 0.0320898i \(0.0102162\pi\)
−0.684052 + 0.729434i \(0.739784\pi\)
\(348\) 0 0
\(349\) 6.41003 + 2.65512i 0.343121 + 0.142125i 0.547588 0.836748i \(-0.315546\pi\)
−0.204467 + 0.978873i \(0.565546\pi\)
\(350\) 0 0
\(351\) −13.8059 6.97115i −0.736905 0.372092i
\(352\) 0 0
\(353\) 10.7742i 0.573453i 0.958012 + 0.286726i \(0.0925671\pi\)
−0.958012 + 0.286726i \(0.907433\pi\)
\(354\) 0 0
\(355\) 6.72880 16.2448i 0.357128 0.862182i
\(356\) 0 0
\(357\) 1.79245 + 7.42630i 0.0948665 + 0.393041i
\(358\) 0 0
\(359\) −7.99933 + 7.99933i −0.422189 + 0.422189i −0.885957 0.463768i \(-0.846497\pi\)
0.463768 + 0.885957i \(0.346497\pi\)
\(360\) 0 0
\(361\) −13.4298 13.4298i −0.706832 0.706832i
\(362\) 0 0
\(363\) 12.5184 3.02151i 0.657046 0.158588i
\(364\) 0 0
\(365\) 6.33256 + 2.62303i 0.331461 + 0.137296i
\(366\) 0 0
\(367\) 4.78544 0.249798 0.124899 0.992169i \(-0.460139\pi\)
0.124899 + 0.992169i \(0.460139\pi\)
\(368\) 0 0
\(369\) 19.9313 + 6.42253i 1.03758 + 0.334343i
\(370\) 0 0
\(371\) 0.806641 1.94740i 0.0418787 0.101104i
\(372\) 0 0
\(373\) 24.5505 10.1691i 1.27118 0.526538i 0.357855 0.933777i \(-0.383508\pi\)
0.913322 + 0.407239i \(0.133508\pi\)
\(374\) 0 0
\(375\) −3.05768 + 2.22722i −0.157898 + 0.115013i
\(376\) 0 0
\(377\) −2.86220 2.86220i −0.147411 0.147411i
\(378\) 0 0
\(379\) 26.9921 11.1805i 1.38649 0.574304i 0.440282 0.897859i \(-0.354878\pi\)
0.946209 + 0.323556i \(0.104878\pi\)
\(380\) 0 0
\(381\) 6.33272 10.3623i 0.324435 0.530878i
\(382\) 0 0
\(383\) −16.7272 −0.854721 −0.427360 0.904081i \(-0.640556\pi\)
−0.427360 + 0.904081i \(0.640556\pi\)
\(384\) 0 0
\(385\) −10.0621 −0.512811
\(386\) 0 0
\(387\) 17.8022 20.9466i 0.904937 1.06478i
\(388\) 0 0
\(389\) 30.3193 12.5587i 1.53725 0.636749i 0.556294 0.830985i \(-0.312223\pi\)
0.980955 + 0.194236i \(0.0622228\pi\)
\(390\) 0 0
\(391\) 8.39850 + 8.39850i 0.424730 + 0.424730i
\(392\) 0 0
\(393\) 4.51893 + 6.20390i 0.227950 + 0.312945i
\(394\) 0 0
\(395\) −1.79448 + 0.743298i −0.0902901 + 0.0373994i
\(396\) 0 0
\(397\) −8.13703 + 19.6445i −0.408386 + 0.985931i 0.577177 + 0.816619i \(0.304154\pi\)
−0.985563 + 0.169312i \(0.945846\pi\)
\(398\) 0 0
\(399\) −0.0177556 + 0.112994i −0.000888894 + 0.00565676i
\(400\) 0 0
\(401\) −0.509396 −0.0254380 −0.0127190 0.999919i \(-0.504049\pi\)
−0.0127190 + 0.999919i \(0.504049\pi\)
\(402\) 0 0
\(403\) 15.5113 + 6.42500i 0.772674 + 0.320052i
\(404\) 0 0
\(405\) −6.27166 + 26.6946i −0.311641 + 1.32647i
\(406\) 0 0
\(407\) 33.2914 + 33.2914i 1.65019 + 1.65019i
\(408\) 0 0
\(409\) −6.54063 + 6.54063i −0.323413 + 0.323413i −0.850075 0.526662i \(-0.823443\pi\)
0.526662 + 0.850075i \(0.323443\pi\)
\(410\) 0 0
\(411\) −14.3930 + 3.47396i −0.709952 + 0.171358i
\(412\) 0 0
\(413\) 0.383497 0.925844i 0.0188707 0.0455578i
\(414\) 0 0
\(415\) 15.4880i 0.760278i
\(416\) 0 0
\(417\) 2.68578 17.0918i 0.131523 0.836990i
\(418\) 0 0
\(419\) 20.7965 + 8.61418i 1.01597 + 0.420830i 0.827631 0.561273i \(-0.189688\pi\)
0.188343 + 0.982103i \(0.439688\pi\)
\(420\) 0 0
\(421\) 8.30131 + 20.0411i 0.404581 + 0.976745i 0.986539 + 0.163526i \(0.0522868\pi\)
−0.581958 + 0.813219i \(0.697713\pi\)
\(422\) 0 0
\(423\) −23.9980 + 12.3012i −1.16682 + 0.598105i
\(424\) 0 0
\(425\) 17.3677 17.3677i 0.842456 0.842456i
\(426\) 0 0
\(427\) 1.17675 + 2.84092i 0.0569468 + 0.137482i
\(428\) 0 0
\(429\) 11.5426 18.8873i 0.557282 0.911889i
\(430\) 0 0
\(431\) 1.97237i 0.0950057i −0.998871 0.0475028i \(-0.984874\pi\)
0.998871 0.0475028i \(-0.0151263\pi\)
\(432\) 0 0
\(433\) 29.4876i 1.41709i 0.705668 + 0.708543i \(0.250647\pi\)
−0.705668 + 0.708543i \(0.749353\pi\)
\(434\) 0 0
\(435\) −3.74238 + 6.12370i −0.179433 + 0.293609i
\(436\) 0 0
\(437\) 0.0680518 + 0.164292i 0.00325536 + 0.00785913i
\(438\) 0 0
\(439\) 17.1823 17.1823i 0.820067 0.820067i −0.166050 0.986117i \(-0.553101\pi\)
0.986117 + 0.166050i \(0.0531014\pi\)
\(440\) 0 0
\(441\) 17.1085 8.76968i 0.814690 0.417604i
\(442\) 0 0
\(443\) −4.22858 10.2087i −0.200906 0.485029i 0.791029 0.611779i \(-0.209546\pi\)
−0.991935 + 0.126749i \(0.959546\pi\)
\(444\) 0 0
\(445\) 3.59309 + 1.48831i 0.170329 + 0.0705526i
\(446\) 0 0
\(447\) −2.83610 + 18.0485i −0.134143 + 0.853663i
\(448\) 0 0
\(449\) 27.4116i 1.29363i −0.762645 0.646817i \(-0.776100\pi\)
0.762645 0.646817i \(-0.223900\pi\)
\(450\) 0 0
\(451\) −11.4691 + 27.6888i −0.540057 + 1.30381i
\(452\) 0 0
\(453\) 19.0227 4.59143i 0.893766 0.215724i
\(454\) 0 0
\(455\) −4.93230 + 4.93230i −0.231230 + 0.231230i
\(456\) 0 0
\(457\) 5.05963 + 5.05963i 0.236680 + 0.236680i 0.815474 0.578794i \(-0.196477\pi\)
−0.578794 + 0.815474i \(0.696477\pi\)
\(458\) 0 0
\(459\) −2.22947 + 29.7134i −0.104063 + 1.38690i
\(460\) 0 0
\(461\) −28.5768 11.8369i −1.33095 0.551298i −0.400025 0.916504i \(-0.630999\pi\)
−0.930927 + 0.365206i \(0.880999\pi\)
\(462\) 0 0
\(463\) −8.48410 −0.394290 −0.197145 0.980374i \(-0.563167\pi\)
−0.197145 + 0.980374i \(0.563167\pi\)
\(464\) 0 0
\(465\) 4.62093 29.4067i 0.214290 1.36371i
\(466\) 0 0
\(467\) −6.00576 + 14.4992i −0.277913 + 0.670942i −0.999778 0.0210925i \(-0.993286\pi\)
0.721864 + 0.692035i \(0.243286\pi\)
\(468\) 0 0
\(469\) −6.23089 + 2.58092i −0.287716 + 0.119176i
\(470\) 0 0
\(471\) −2.93003 4.02254i −0.135008 0.185349i
\(472\) 0 0
\(473\) 27.8199 + 27.8199i 1.27916 + 1.27916i
\(474\) 0 0
\(475\) 0.339747 0.140728i 0.0155886 0.00645703i
\(476\) 0 0
\(477\) 5.32414 6.26455i 0.243776 0.286834i
\(478\) 0 0
\(479\) 28.2696 1.29167 0.645835 0.763477i \(-0.276509\pi\)
0.645835 + 0.763477i \(0.276509\pi\)
\(480\) 0 0
\(481\) 32.6380 1.48816
\(482\) 0 0
\(483\) −1.43889 + 2.35447i −0.0654716 + 0.107132i
\(484\) 0 0
\(485\) −37.4533 + 15.5137i −1.70067 + 0.704440i
\(486\) 0 0
\(487\) 22.4971 + 22.4971i 1.01944 + 1.01944i 0.999807 + 0.0196335i \(0.00624994\pi\)
0.0196335 + 0.999807i \(0.493750\pi\)
\(488\) 0 0
\(489\) 12.8712 9.37542i 0.582056 0.423971i
\(490\) 0 0
\(491\) −0.871501 + 0.360988i −0.0393303 + 0.0162911i −0.402262 0.915525i \(-0.631776\pi\)
0.362932 + 0.931816i \(0.381776\pi\)
\(492\) 0 0
\(493\) −2.98432 + 7.20480i −0.134407 + 0.324488i
\(494\) 0 0
\(495\) −37.3542 12.0368i −1.67895 0.541013i
\(496\) 0 0
\(497\) 4.43881 0.199108
\(498\) 0 0
\(499\) −29.0833 12.0467i −1.30195 0.539284i −0.379423 0.925223i \(-0.623878\pi\)
−0.922524 + 0.385939i \(0.873878\pi\)
\(500\) 0 0
\(501\) 15.9633 3.85299i 0.713188 0.172139i
\(502\) 0 0
\(503\) 15.9266 + 15.9266i 0.710132 + 0.710132i 0.966563 0.256431i \(-0.0825465\pi\)
−0.256431 + 0.966563i \(0.582546\pi\)
\(504\) 0 0
\(505\) −12.8126 + 12.8126i −0.570152 + 0.570152i
\(506\) 0 0
\(507\) 1.68274 + 6.97176i 0.0747331 + 0.309627i
\(508\) 0 0
\(509\) 0.374204 0.903408i 0.0165863 0.0400428i −0.915369 0.402615i \(-0.868101\pi\)
0.931956 + 0.362572i \(0.118101\pi\)
\(510\) 0 0
\(511\) 1.73035i 0.0765460i
\(512\) 0 0
\(513\) −0.201085 + 0.398235i −0.00887810 + 0.0175825i
\(514\) 0 0
\(515\) −31.3239 12.9748i −1.38030 0.571738i
\(516\) 0 0
\(517\) −14.7698 35.6575i −0.649576 1.56821i
\(518\) 0 0
\(519\) −20.6553 28.3570i −0.906666 1.24473i
\(520\) 0 0
\(521\) −31.7757 + 31.7757i −1.39212 + 1.39212i −0.571558 + 0.820562i \(0.693661\pi\)
−0.820562 + 0.571558i \(0.806339\pi\)
\(522\) 0 0
\(523\) −2.56429 6.19074i −0.112128 0.270702i 0.857847 0.513905i \(-0.171802\pi\)
−0.969975 + 0.243203i \(0.921802\pi\)
\(524\) 0 0
\(525\) 4.86893 + 2.97554i 0.212497 + 0.129863i
\(526\) 0 0
\(527\) 32.3463i 1.40903i
\(528\) 0 0
\(529\) 18.7100i 0.813480i
\(530\) 0 0
\(531\) 2.53123 2.97832i 0.109846 0.129248i
\(532\) 0 0
\(533\) 7.95069 + 19.1947i 0.344382 + 0.831413i
\(534\) 0 0
\(535\) −2.10224 + 2.10224i −0.0908876 + 0.0908876i
\(536\) 0 0
\(537\) 25.6130 18.6566i 1.10528 0.805090i
\(538\) 0 0
\(539\) 10.5296 + 25.4206i 0.453541 + 1.09494i
\(540\) 0 0
\(541\) −25.9983 10.7689i −1.11776 0.462989i −0.254154 0.967164i \(-0.581797\pi\)
−0.863602 + 0.504174i \(0.831797\pi\)
\(542\) 0 0
\(543\) −26.8373 4.21717i −1.15170 0.180976i
\(544\) 0 0
\(545\) 3.64369i 0.156079i
\(546\) 0 0
\(547\) −8.74735 + 21.1180i −0.374010 + 0.902939i 0.619053 + 0.785349i \(0.287517\pi\)
−0.993062 + 0.117590i \(0.962483\pi\)
\(548\) 0 0
\(549\) 0.970075 + 11.9542i 0.0414018 + 0.510195i
\(550\) 0 0
\(551\) −0.0825609 + 0.0825609i −0.00351721 + 0.00351721i
\(552\) 0 0
\(553\) −0.346719 0.346719i −0.0147440 0.0147440i
\(554\) 0 0
\(555\) −13.5773 56.2520i −0.576323 2.38777i
\(556\) 0 0
\(557\) 21.4222 + 8.87338i 0.907689 + 0.375977i 0.787171 0.616734i \(-0.211545\pi\)
0.120517 + 0.992711i \(0.461545\pi\)
\(558\) 0 0
\(559\) 27.2739 1.15356
\(560\) 0 0
\(561\) −42.1285 6.62001i −1.77867 0.279497i
\(562\) 0 0
\(563\) −3.72042 + 8.98190i −0.156797 + 0.378542i −0.982683 0.185296i \(-0.940676\pi\)
0.825886 + 0.563838i \(0.190676\pi\)
\(564\) 0 0
\(565\) −23.8830 + 9.89265i −1.00476 + 0.416187i
\(566\) 0 0
\(567\) −6.83187 + 1.11615i −0.286912 + 0.0468739i
\(568\) 0 0
\(569\) −8.43864 8.43864i −0.353766 0.353766i 0.507743 0.861509i \(-0.330480\pi\)
−0.861509 + 0.507743i \(0.830480\pi\)
\(570\) 0 0
\(571\) −17.9095 + 7.41834i −0.749488 + 0.310448i −0.724532 0.689241i \(-0.757944\pi\)
−0.0249551 + 0.999689i \(0.507944\pi\)
\(572\) 0 0
\(573\) −23.2495 14.2085i −0.971264 0.593568i
\(574\) 0 0
\(575\) 8.87142 0.369964
\(576\) 0 0
\(577\) −31.3503 −1.30513 −0.652565 0.757733i \(-0.726307\pi\)
−0.652565 + 0.757733i \(0.726307\pi\)
\(578\) 0 0
\(579\) 18.0672 + 11.0414i 0.750849 + 0.458866i
\(580\) 0 0
\(581\) −3.61227 + 1.49625i −0.149862 + 0.0620750i
\(582\) 0 0
\(583\) 8.32014 + 8.32014i 0.344585 + 0.344585i
\(584\) 0 0
\(585\) −24.2108 + 12.4103i −1.00099 + 0.513102i
\(586\) 0 0
\(587\) 14.6871 6.08361i 0.606202 0.251097i −0.0584014 0.998293i \(-0.518600\pi\)
0.664604 + 0.747196i \(0.268600\pi\)
\(588\) 0 0
\(589\) 0.185331 0.447428i 0.00763642 0.0184360i
\(590\) 0 0
\(591\) 39.9944 + 6.28465i 1.64515 + 0.258516i
\(592\) 0 0
\(593\) −6.53460 −0.268344 −0.134172 0.990958i \(-0.542837\pi\)
−0.134172 + 0.990958i \(0.542837\pi\)
\(594\) 0 0
\(595\) 12.4157 + 5.14275i 0.508994 + 0.210832i
\(596\) 0 0
\(597\) 0.906779 + 3.75688i 0.0371120 + 0.153759i
\(598\) 0 0
\(599\) −20.5360 20.5360i −0.839078 0.839078i 0.149659 0.988738i \(-0.452182\pi\)
−0.988738 + 0.149659i \(0.952182\pi\)
\(600\) 0 0
\(601\) −1.20249 + 1.20249i −0.0490506 + 0.0490506i −0.731207 0.682156i \(-0.761042\pi\)
0.682156 + 0.731207i \(0.261042\pi\)
\(602\) 0 0
\(603\) −26.2189 + 2.12763i −1.06771 + 0.0866440i
\(604\) 0 0
\(605\) 8.66907 20.9290i 0.352448 0.850884i
\(606\) 0 0
\(607\) 42.2470i 1.71475i −0.514692 0.857375i \(-0.672094\pi\)
0.514692 0.857375i \(-0.327906\pi\)
\(608\) 0 0
\(609\) −1.78977 0.281241i −0.0725251 0.0113965i
\(610\) 0 0
\(611\) −24.7188 10.2389i −1.00002 0.414220i
\(612\) 0 0
\(613\) 3.20549 + 7.73873i 0.129468 + 0.312564i 0.975300 0.220886i \(-0.0708950\pi\)
−0.845831 + 0.533451i \(0.820895\pi\)
\(614\) 0 0
\(615\) 29.7748 21.6880i 1.20064 0.874545i
\(616\) 0 0
\(617\) 30.2757 30.2757i 1.21885 1.21885i 0.250821 0.968034i \(-0.419299\pi\)
0.968034 0.250821i \(-0.0807005\pi\)
\(618\) 0 0
\(619\) 7.03045 + 16.9730i 0.282578 + 0.682203i 0.999894 0.0145446i \(-0.00462985\pi\)
−0.717317 + 0.696747i \(0.754630\pi\)
\(620\) 0 0
\(621\) −8.15823 + 7.01941i −0.327379 + 0.281679i
\(622\) 0 0
\(623\) 0.981797i 0.0393349i
\(624\) 0 0
\(625\) 28.0703i 1.12281i
\(626\) 0 0
\(627\) −0.544810 0.332949i −0.0217576 0.0132967i
\(628\) 0 0
\(629\) −24.0633 58.0939i −0.959465 2.31635i
\(630\) 0 0
\(631\) −0.699961 + 0.699961i −0.0278650 + 0.0278650i −0.720902 0.693037i \(-0.756272\pi\)
0.693037 + 0.720902i \(0.256272\pi\)
\(632\) 0 0
\(633\) −27.9860 38.4212i −1.11235 1.52710i
\(634\) 0 0
\(635\) −8.17515 19.7366i −0.324421 0.783221i
\(636\) 0 0
\(637\) 17.6223 + 7.29941i 0.698222 + 0.289213i
\(638\) 0 0
\(639\) 16.4785 + 5.30993i 0.651881 + 0.210058i
\(640\) 0 0
\(641\) 36.4715i 1.44054i 0.693695 + 0.720269i \(0.255981\pi\)
−0.693695 + 0.720269i \(0.744019\pi\)
\(642\) 0 0
\(643\) 13.4470 32.4640i 0.530298 1.28025i −0.401027 0.916066i \(-0.631347\pi\)
0.931326 0.364187i \(-0.118653\pi\)
\(644\) 0 0
\(645\) −11.3458 47.0069i −0.446741 1.85089i
\(646\) 0 0
\(647\) −7.01510 + 7.01510i −0.275792 + 0.275792i −0.831427 0.555635i \(-0.812475\pi\)
0.555635 + 0.831427i \(0.312475\pi\)
\(648\) 0 0
\(649\) 3.95560 + 3.95560i 0.155271 + 0.155271i
\(650\) 0 0
\(651\) 7.30494 1.76316i 0.286303 0.0691036i
\(652\) 0 0
\(653\) −7.33264 3.03728i −0.286948 0.118858i 0.234567 0.972100i \(-0.424633\pi\)
−0.521515 + 0.853242i \(0.674633\pi\)
\(654\) 0 0
\(655\) 13.5014 0.527544
\(656\) 0 0
\(657\) −2.06993 + 6.42370i −0.0807556 + 0.250612i
\(658\) 0 0
\(659\) 9.46381 22.8477i 0.368658 0.890018i −0.625313 0.780374i \(-0.715029\pi\)
0.993971 0.109644i \(-0.0349712\pi\)
\(660\) 0 0
\(661\) 4.59417 1.90297i 0.178693 0.0740169i −0.291543 0.956558i \(-0.594169\pi\)
0.470236 + 0.882541i \(0.344169\pi\)
\(662\) 0 0
\(663\) −23.8959 + 17.4058i −0.928039 + 0.675986i
\(664\) 0 0
\(665\) 0.142273 + 0.142273i 0.00551713 + 0.00551713i
\(666\) 0 0
\(667\) −2.60230 + 1.07791i −0.100762 + 0.0417368i
\(668\) 0 0
\(669\) −21.3395 + 34.9181i −0.825033 + 1.35001i
\(670\) 0 0
\(671\) −17.1652 −0.662654
\(672\) 0 0
\(673\) −14.0193 −0.540404 −0.270202 0.962804i \(-0.587090\pi\)
−0.270202 + 0.962804i \(0.587090\pi\)
\(674\) 0 0
\(675\) 14.5158 + 16.8708i 0.558713 + 0.649358i
\(676\) 0 0
\(677\) −25.4610 + 10.5463i −0.978545 + 0.405327i −0.813886 0.581024i \(-0.802652\pi\)
−0.164659 + 0.986351i \(0.552652\pi\)
\(678\) 0 0
\(679\) −7.23650 7.23650i −0.277712 0.277712i
\(680\) 0 0
\(681\) 18.4836 + 25.3756i 0.708294 + 0.972395i
\(682\) 0 0
\(683\) −7.18630 + 2.97666i −0.274976 + 0.113899i −0.515910 0.856643i \(-0.672546\pi\)
0.240934 + 0.970541i \(0.422546\pi\)
\(684\) 0 0
\(685\) −9.96720 + 24.0630i −0.380827 + 0.919398i
\(686\) 0 0
\(687\) −3.55268 + 22.6086i −0.135543 + 0.862573i
\(688\) 0 0
\(689\) 8.15685 0.310751
\(690\) 0 0
\(691\) 21.5412 + 8.92265i 0.819466 + 0.339434i 0.752724 0.658336i \(-0.228740\pi\)
0.0667419 + 0.997770i \(0.478740\pi\)
\(692\) 0 0
\(693\) −0.801343 9.87496i −0.0304405 0.375119i
\(694\) 0 0
\(695\) −21.5208 21.5208i −0.816330 0.816330i
\(696\) 0 0
\(697\) 28.3036 28.3036i 1.07207 1.07207i
\(698\) 0 0
\(699\) 18.8911 4.55965i 0.714527 0.172462i
\(700\) 0 0
\(701\) −12.8390 + 30.9961i −0.484922 + 1.17071i 0.472322 + 0.881426i \(0.343416\pi\)
−0.957244 + 0.289280i \(0.906584\pi\)
\(702\) 0 0
\(703\) 0.941452i 0.0355075i
\(704\) 0 0
\(705\) −7.36389 + 46.8625i −0.277340 + 1.76494i
\(706\) 0 0
\(707\) −4.22606 1.75049i −0.158937 0.0658339i
\(708\) 0 0
\(709\) 0.0401422 + 0.0969118i 0.00150757 + 0.00363960i 0.924632 0.380863i \(-0.124373\pi\)
−0.923124 + 0.384502i \(0.874373\pi\)
\(710\) 0 0
\(711\) −0.872388 1.70191i −0.0327171 0.0638267i
\(712\) 0 0
\(713\) 8.26126 8.26126i 0.309387 0.309387i
\(714\) 0 0
\(715\) −14.9008 35.9737i −0.557258 1.34534i
\(716\) 0 0
\(717\) −15.1068 + 24.7195i −0.564175 + 0.923168i
\(718\) 0 0
\(719\) 7.25362i 0.270515i 0.990811 + 0.135257i \(0.0431861\pi\)
−0.990811 + 0.135257i \(0.956814\pi\)
\(720\) 0 0
\(721\) 8.55913i 0.318759i
\(722\) 0 0
\(723\) 8.13075 13.3045i 0.302386 0.494798i
\(724\) 0 0
\(725\) 2.22906 + 5.38143i 0.0827853 + 0.199861i
\(726\) 0 0
\(727\) −33.8662 + 33.8662i −1.25603 + 1.25603i −0.303055 + 0.952973i \(0.598007\pi\)
−0.952973 + 0.303055i \(0.901993\pi\)
\(728\) 0 0
\(729\) −26.6977 4.02907i −0.988803 0.149225i
\(730\) 0 0
\(731\) −20.1084 48.5460i −0.743736 1.79554i
\(732\) 0 0
\(733\) 46.5930 + 19.2995i 1.72095 + 0.712842i 0.999799 + 0.0200663i \(0.00638774\pi\)
0.721153 + 0.692775i \(0.243612\pi\)
\(734\) 0 0
\(735\) 5.24981 33.4088i 0.193642 1.23230i
\(736\) 0 0
\(737\) 37.6478i 1.38678i
\(738\) 0 0
\(739\) −11.4353 + 27.6072i −0.420653 + 1.01555i 0.561502 + 0.827475i \(0.310224\pi\)
−0.982155 + 0.188072i \(0.939776\pi\)
\(740\) 0 0
\(741\) −0.430266 + 0.103851i −0.0158062 + 0.00381507i
\(742\) 0 0
\(743\) 31.4330 31.4330i 1.15317 1.15317i 0.167252 0.985914i \(-0.446511\pi\)
0.985914 0.167252i \(-0.0534894\pi\)
\(744\) 0 0
\(745\) 22.7253 + 22.7253i 0.832590 + 0.832590i
\(746\) 0 0
\(747\) −15.2000 + 1.23347i −0.556139 + 0.0451301i
\(748\) 0 0
\(749\) −0.693395 0.287214i −0.0253361 0.0104946i
\(750\) 0 0
\(751\) 9.38556 0.342484 0.171242 0.985229i \(-0.445222\pi\)
0.171242 + 0.985229i \(0.445222\pi\)
\(752\) 0 0
\(753\) 0.422577 2.68920i 0.0153996 0.0980001i
\(754\) 0 0
\(755\) 13.1733 31.8033i 0.479427 1.15744i
\(756\) 0 0
\(757\) 21.2091 8.78508i 0.770857 0.319299i 0.0376373 0.999291i \(-0.488017\pi\)
0.733219 + 0.679992i \(0.238017\pi\)
\(758\) 0 0
\(759\) −9.06889 12.4504i −0.329180 0.451921i
\(760\) 0 0
\(761\) −17.8648 17.8648i −0.647599 0.647599i 0.304813 0.952412i \(-0.401406\pi\)
−0.952412 + 0.304813i \(0.901406\pi\)
\(762\) 0 0
\(763\) −0.849818 + 0.352006i −0.0307655 + 0.0127435i
\(764\) 0 0
\(765\) 39.9397 + 33.9441i 1.44402 + 1.22725i
\(766\) 0 0
\(767\) 3.87797 0.140025
\(768\) 0 0
\(769\) −1.71307 −0.0617750 −0.0308875 0.999523i \(-0.509833\pi\)
−0.0308875 + 0.999523i \(0.509833\pi\)
\(770\) 0 0
\(771\) −7.48697 + 12.2510i −0.269637 + 0.441210i
\(772\) 0 0
\(773\) −25.0667 + 10.3830i −0.901587 + 0.373450i −0.784830 0.619711i \(-0.787250\pi\)
−0.116757 + 0.993161i \(0.537250\pi\)
\(774\) 0 0
\(775\) −17.0839 17.0839i −0.613671 0.613671i
\(776\) 0 0
\(777\) 11.8080 8.60096i 0.423609 0.308558i
\(778\) 0 0
\(779\) 0.553675 0.229340i 0.0198375 0.00821694i
\(780\) 0 0
\(781\) −9.48225 + 22.8922i −0.339302 + 0.819146i
\(782\) 0 0
\(783\) −6.30787 3.18509i −0.225425 0.113826i
\(784\) 0 0
\(785\) −8.75417 −0.312450
\(786\) 0 0
\(787\) 47.6483 + 19.7366i 1.69848 + 0.703533i 0.999928 0.0120263i \(-0.00382819\pi\)
0.698552 + 0.715560i \(0.253828\pi\)
\(788\) 0 0
\(789\) 20.8616 5.03525i 0.742691 0.179260i
\(790\) 0 0
\(791\) −4.61452 4.61452i −0.164073 0.164073i
\(792\) 0 0
\(793\) −8.41414 + 8.41414i −0.298795 + 0.298795i
\(794\) 0 0
\(795\) −3.39322 14.0584i −0.120345 0.498601i
\(796\) 0 0
\(797\) 9.69549 23.4070i 0.343432 0.829118i −0.653932 0.756553i \(-0.726882\pi\)
0.997364 0.0725644i \(-0.0231183\pi\)
\(798\) 0 0
\(799\) 51.5470i 1.82360i
\(800\) 0 0
\(801\) −1.17448 + 3.64480i −0.0414981 + 0.128783i
\(802\) 0 0
\(803\) −8.92387 3.69639i −0.314917 0.130443i
\(804\) 0 0
\(805\) 1.85751 + 4.48443i 0.0654687 + 0.158055i
\(806\) 0 0
\(807\) 16.6699 + 22.8856i 0.586808 + 0.805611i
\(808\) 0 0
\(809\) −31.1610 + 31.1610i −1.09556 + 1.09556i −0.100640 + 0.994923i \(0.532089\pi\)
−0.994923 + 0.100640i \(0.967911\pi\)
\(810\) 0 0
\(811\) 3.17453 + 7.66400i 0.111473 + 0.269120i 0.969764 0.244043i \(-0.0784738\pi\)
−0.858291 + 0.513163i \(0.828474\pi\)
\(812\) 0 0
\(813\) −25.8687 15.8091i −0.907254 0.554450i
\(814\) 0 0
\(815\) 28.0113i 0.981194i
\(816\) 0 0
\(817\) 0.786722i 0.0275239i
\(818\) 0 0
\(819\) −5.23338 4.44777i −0.182869 0.155418i
\(820\) 0 0
\(821\) −1.18631 2.86400i −0.0414025 0.0999544i 0.901826 0.432100i \(-0.142227\pi\)
−0.943228 + 0.332146i \(0.892227\pi\)
\(822\) 0 0
\(823\) −1.98030 + 1.98030i −0.0690288 + 0.0690288i −0.740778 0.671750i \(-0.765543\pi\)
0.671750 + 0.740778i \(0.265543\pi\)
\(824\) 0 0
\(825\) −25.7468 + 18.7540i −0.896388 + 0.652931i
\(826\) 0 0
\(827\) 19.1750 + 46.2924i 0.666779 + 1.60975i 0.786967 + 0.616995i \(0.211650\pi\)
−0.120189 + 0.992751i \(0.538350\pi\)
\(828\) 0 0
\(829\) −20.6566 8.55623i −0.717432 0.297170i −0.00605590 0.999982i \(-0.501928\pi\)
−0.711376 + 0.702812i \(0.751928\pi\)
\(830\) 0 0
\(831\) −39.5750 6.21875i −1.37284 0.215726i
\(832\) 0 0
\(833\) 36.7485i 1.27326i
\(834\) 0 0
\(835\) 11.0547 26.6884i 0.382563 0.923589i
\(836\) 0 0
\(837\) 29.2279 + 2.19304i 1.01026 + 0.0758026i
\(838\) 0 0
\(839\) −10.0632 + 10.0632i −0.347420 + 0.347420i −0.859148 0.511728i \(-0.829006\pi\)
0.511728 + 0.859148i \(0.329006\pi\)
\(840\) 0 0
\(841\) 19.1984 + 19.1984i 0.662013 + 0.662013i
\(842\) 0 0
\(843\) 3.41059 + 14.1304i 0.117467 + 0.486678i
\(844\) 0 0
\(845\) 11.6558 + 4.82798i 0.400971 + 0.166088i
\(846\) 0 0
\(847\) 5.71875 0.196499
\(848\) 0 0
\(849\) 43.9904 + 6.91258i 1.50975 + 0.237239i
\(850\) 0 0
\(851\) 8.69142 20.9830i 0.297938 0.719286i
\(852\) 0 0
\(853\) 11.2850 4.67440i 0.386391 0.160048i −0.181027 0.983478i \(-0.557942\pi\)
0.567418 + 0.823430i \(0.307942\pi\)
\(854\) 0 0
\(855\) 0.357978 + 0.698368i 0.0122426 + 0.0238837i
\(856\) 0 0
\(857\) 22.7269 + 22.7269i 0.776335 + 0.776335i 0.979206 0.202871i \(-0.0650271\pi\)
−0.202871 + 0.979206i \(0.565027\pi\)
\(858\) 0 0
\(859\) 11.7509 4.86736i 0.400934 0.166072i −0.173099 0.984904i \(-0.555378\pi\)
0.574033 + 0.818832i \(0.305378\pi\)
\(860\) 0 0
\(861\) 7.93474 + 4.84916i 0.270415 + 0.165259i
\(862\) 0 0
\(863\) 45.0560 1.53372 0.766862 0.641812i \(-0.221817\pi\)
0.766862 + 0.641812i \(0.221817\pi\)
\(864\) 0 0
\(865\) −61.7127 −2.09829
\(866\) 0 0
\(867\) 23.4747 + 14.3461i 0.797244 + 0.487219i
\(868\) 0 0
\(869\) 2.52879 1.04746i 0.0857833 0.0355326i
\(870\) 0 0
\(871\) −18.4545 18.4545i −0.625306 0.625306i
\(872\) 0 0
\(873\) −18.2080 35.5213i −0.616246 1.20221i
\(874\) 0 0
\(875\) −1.55199 + 0.642857i −0.0524669 + 0.0217325i
\(876\) 0 0
\(877\) 6.55184 15.8175i 0.221240 0.534121i −0.773819 0.633407i \(-0.781656\pi\)
0.995059 + 0.0992863i \(0.0316560\pi\)
\(878\) 0 0
\(879\) 7.77914 + 1.22240i 0.262384 + 0.0412305i
\(880\) 0 0
\(881\) −20.4889 −0.690289 −0.345144 0.938550i \(-0.612170\pi\)
−0.345144 + 0.938550i \(0.612170\pi\)
\(882\) 0 0
\(883\) −3.85593 1.59718i −0.129762 0.0537493i 0.316857 0.948473i \(-0.397372\pi\)
−0.446620 + 0.894724i \(0.647372\pi\)
\(884\) 0 0
\(885\) −1.61322 6.68373i −0.0542278 0.224671i
\(886\) 0 0
\(887\) 29.9883 + 29.9883i 1.00691 + 1.00691i 0.999976 + 0.00693107i \(0.00220625\pi\)
0.00693107 + 0.999976i \(0.497794\pi\)
\(888\) 0 0
\(889\) 3.81338 3.81338i 0.127897 0.127897i
\(890\) 0 0
\(891\) 8.83805 37.6182i 0.296086 1.26026i
\(892\) 0 0
\(893\) −0.295343 + 0.713020i −0.00988326 + 0.0238603i
\(894\) 0 0
\(895\) 55.7410i 1.86322i
\(896\) 0 0
\(897\) −10.5485 1.65757i −0.352203 0.0553446i
\(898\) 0 0
\(899\) 7.08706 + 2.93556i 0.236367 + 0.0979063i
\(900\) 0 0
\(901\) −6.01386 14.5187i −0.200351 0.483690i
\(902\) 0 0
\(903\) 9.86732 7.18737i 0.328364 0.239181i
\(904\) 0 0
\(905\) −33.7916 + 33.7916i −1.12327 + 1.12327i
\(906\) 0 0
\(907\) −16.6011 40.0785i −0.551229 1.33078i −0.916557 0.399905i \(-0.869043\pi\)
0.365328 0.930879i \(-0.380957\pi\)
\(908\) 0 0
\(909\) −13.5947 11.5539i −0.450907 0.383219i
\(910\) 0 0
\(911\) 42.9300i 1.42233i −0.703023 0.711167i \(-0.748167\pi\)
0.703023 0.711167i \(-0.251833\pi\)
\(912\) 0 0
\(913\) 21.8258i 0.722328i
\(914\) 0 0
\(915\) 18.0021 + 11.0016i 0.595132 + 0.363703i
\(916\) 0 0
\(917\) 1.30433 + 3.14893i 0.0430728 + 0.103987i
\(918\) 0 0
\(919\) −27.6253 + 27.6253i −0.911275 + 0.911275i −0.996373 0.0850978i \(-0.972880\pi\)
0.0850978 + 0.996373i \(0.472880\pi\)
\(920\) 0 0
\(921\) 13.6479 + 18.7368i 0.449714 + 0.617398i
\(922\) 0 0
\(923\) 6.57337 + 15.8695i 0.216365 + 0.522351i
\(924\) 0 0
\(925\) −43.3917 17.9734i −1.42671 0.590963i
\(926\) 0 0
\(927\) 10.2389 31.7747i 0.336289 1.04362i
\(928\) 0 0
\(929\) 26.5634i 0.871518i 0.900063 + 0.435759i \(0.143520\pi\)
−0.900063 + 0.435759i \(0.856480\pi\)
\(930\) 0 0
\(931\) 0.210553 0.508321i 0.00690060 0.0166595i
\(932\) 0 0
\(933\) −6.16692 25.5502i −0.201896 0.836475i
\(934\) 0 0
\(935\) −53.0452 + 53.0452i −1.73476 + 1.73476i
\(936\) 0 0
\(937\) −1.34624 1.34624i −0.0439798 0.0439798i 0.684775 0.728755i \(-0.259901\pi\)
−0.728755 + 0.684775i \(0.759901\pi\)
\(938\) 0 0
\(939\) −42.6150 + 10.2858i −1.39069 + 0.335663i
\(940\) 0 0
\(941\) 6.86683 + 2.84433i 0.223852 + 0.0927226i 0.491791 0.870713i \(-0.336342\pi\)
−0.267939 + 0.963436i \(0.586342\pi\)
\(942\) 0 0
\(943\) 14.4575 0.470801
\(944\) 0 0
\(945\) −5.48872 + 10.8701i −0.178548 + 0.353603i
\(946\) 0 0
\(947\) −13.0476 + 31.4996i −0.423989 + 1.02360i 0.557170 + 0.830398i \(0.311887\pi\)
−0.981159 + 0.193201i \(0.938113\pi\)
\(948\) 0 0
\(949\) −6.18628 + 2.56244i −0.200815 + 0.0831804i
\(950\) 0 0
\(951\) 9.50001 6.91983i 0.308059 0.224391i
\(952\) 0 0
\(953\) 1.94423 + 1.94423i 0.0629798 + 0.0629798i 0.737895 0.674915i \(-0.235820\pi\)
−0.674915 + 0.737895i \(0.735820\pi\)
\(954\) 0 0
\(955\) −44.2822 + 18.3423i −1.43294 + 0.593542i
\(956\) 0 0
\(957\) 5.27377 8.62955i 0.170477 0.278954i
\(958\) 0 0
\(959\) −6.57510 −0.212321
\(960\) 0 0
\(961\) −0.817748 −0.0263790
\(962\) 0 0
\(963\) −2.23056 1.89572i −0.0718789 0.0610887i
\(964\) 0 0
\(965\) 34.4117 14.2538i 1.10775 0.458846i
\(966\) 0 0
\(967\) −30.8641 30.8641i −0.992522 0.992522i 0.00745003 0.999972i \(-0.497629\pi\)
−0.999972 + 0.00745003i \(0.997629\pi\)
\(968\) 0 0
\(969\) 0.502075 + 0.689283i 0.0161290 + 0.0221430i
\(970\) 0 0
\(971\) −10.5485 + 4.36935i −0.338519 + 0.140219i −0.545466 0.838133i \(-0.683647\pi\)
0.206947 + 0.978352i \(0.433647\pi\)
\(972\) 0 0
\(973\) 2.94023 7.09834i 0.0942594 0.227562i
\(974\) 0 0
\(975\) −3.42777 + 21.8137i −0.109776 + 0.698597i
\(976\) 0 0
\(977\) −19.0100 −0.608184 −0.304092 0.952643i \(-0.598353\pi\)
−0.304092 + 0.952643i \(0.598353\pi\)
\(978\) 0 0
\(979\) −5.06340 2.09733i −0.161827 0.0670309i
\(980\) 0 0
\(981\) −3.57593 + 0.290183i −0.114171 + 0.00926485i
\(982\) 0 0
\(983\) 20.9677 + 20.9677i 0.668767 + 0.668767i 0.957431 0.288664i \(-0.0932110\pi\)
−0.288664 + 0.957431i \(0.593211\pi\)
\(984\) 0 0
\(985\) 50.3580 50.3580i 1.60454 1.60454i
\(986\) 0 0
\(987\) −11.6411 + 2.80976i −0.370541 + 0.0894357i
\(988\) 0 0
\(989\) 7.26297 17.5344i 0.230949 0.557560i
\(990\) 0 0
\(991\) 45.2241i 1.43659i 0.695737 + 0.718296i \(0.255078\pi\)
−0.695737 + 0.718296i \(0.744922\pi\)
\(992\) 0 0
\(993\) −2.81706 + 17.9273i −0.0893968 + 0.568905i
\(994\) 0 0
\(995\) 6.28096 + 2.60166i 0.199120 + 0.0824781i
\(996\) 0 0
\(997\) −20.9163 50.4964i −0.662425 1.59924i −0.793991 0.607929i \(-0.792000\pi\)
0.131566 0.991307i \(-0.458000\pi\)
\(998\) 0 0
\(999\) 54.1246 17.8047i 1.71243 0.563315i
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 768.2.o.a.95.13 56
3.2 odd 2 inner 768.2.o.a.95.2 56
4.3 odd 2 768.2.o.b.95.2 56
8.3 odd 2 96.2.o.a.35.12 yes 56
8.5 even 2 384.2.o.a.47.2 56
12.11 even 2 768.2.o.b.95.13 56
24.5 odd 2 384.2.o.a.47.13 56
24.11 even 2 96.2.o.a.35.3 yes 56
32.5 even 8 96.2.o.a.11.3 56
32.11 odd 8 inner 768.2.o.a.671.2 56
32.21 even 8 768.2.o.b.671.13 56
32.27 odd 8 384.2.o.a.335.13 56
96.5 odd 8 96.2.o.a.11.12 yes 56
96.11 even 8 inner 768.2.o.a.671.13 56
96.53 odd 8 768.2.o.b.671.2 56
96.59 even 8 384.2.o.a.335.2 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.o.a.11.3 56 32.5 even 8
96.2.o.a.11.12 yes 56 96.5 odd 8
96.2.o.a.35.3 yes 56 24.11 even 2
96.2.o.a.35.12 yes 56 8.3 odd 2
384.2.o.a.47.2 56 8.5 even 2
384.2.o.a.47.13 56 24.5 odd 2
384.2.o.a.335.2 56 96.59 even 8
384.2.o.a.335.13 56 32.27 odd 8
768.2.o.a.95.2 56 3.2 odd 2 inner
768.2.o.a.95.13 56 1.1 even 1 trivial
768.2.o.a.671.2 56 32.11 odd 8 inner
768.2.o.a.671.13 56 96.11 even 8 inner
768.2.o.b.95.2 56 4.3 odd 2
768.2.o.b.95.13 56 12.11 even 2
768.2.o.b.671.2 56 96.53 odd 8
768.2.o.b.671.13 56 32.21 even 8