LMFDB - Embedded newform 216.3.e.b.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,3,Mod(161,216)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("216.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	216.e (of order $2$, degree $1$, 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:	$5.88557371018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	216.161
Dual form			216.3.e.b.161.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685i q^{5} +3.00000 q^{7} +O(q^{10})$	$q+5.65685i q^{5} +3.00000 q^{7} +5.65685i q^{11} -17.0000 q^{13} +28.2843i q^{17} +11.0000 q^{19} +39.5980i q^{23} -7.00000 q^{25} -33.9411i q^{29} +50.0000 q^{31} +16.9706i q^{35} -33.0000 q^{37} +33.9411i q^{41} +10.0000 q^{43} -84.8528i q^{47} -40.0000 q^{49} -11.3137i q^{53} -32.0000 q^{55} -28.2843i q^{59} -41.0000 q^{61} -96.1665i q^{65} +83.0000 q^{67} -22.6274i q^{71} +127.000 q^{73} +16.9706i q^{77} +19.0000 q^{79} -124.451i q^{83} -160.000 q^{85} -84.8528i q^{89} -51.0000 q^{91} +62.2254i q^{95} +167.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{7}+O(q^{10}) $ 2 * q + 6 * q^7	$ 2 q + 6 q^{7} - 34 q^{13} + 22 q^{19} - 14 q^{25} + 100 q^{31} - 66 q^{37} + 20 q^{43} - 80 q^{49} - 64 q^{55} - 82 q^{61} + 166 q^{67} + 254 q^{73} + 38 q^{79} - 320 q^{85} - 102 q^{91} + 334 q^{97}+O(q^{100}) $ 2 * q + 6 * q^7 - 34 * q^13 + 22 * q^19 - 14 * q^25 + 100 * q^31 - 66 * q^37 + 20 * q^43 - 80 * q^49 - 64 * q^55 - 82 * q^61 + 166 * q^67 + 254 * q^73 + 38 * q^79 - 320 * q^85 - 102 * q^91 + 334 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.65685i	1.13137i	0.824621	+	0.565685i	$0.191388\pi$
$5$	5.65685i	1.13137i	−0.824621	+	0.565685i	$0.808612\pi$
$6$	0	0
$7$	3.00000	0.428571	0.214286	−	0.976771i	$-0.431258\pi$
$7$	3.00000	0.428571	0.214286	+	0.976771i	$0.431258\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.65685i	0.514259i	0.966377	+	0.257130i	$0.0827768\pi$
$11$	5.65685i	0.514259i	−0.966377	+	0.257130i	$0.917223\pi$
$12$	0	0
$13$	−17.0000	−1.30769	−0.653846	−	0.756628i	$-0.726846\pi$
$13$	−17.0000	−1.30769	−0.653846	+	0.756628i	$0.726846\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	28.2843i	1.66378i	0.554940	+	0.831890i	$0.312741\pi$
$17$	28.2843i	1.66378i	−0.554940	+	0.831890i	$0.687259\pi$
$18$	0	0
$19$	11.0000	0.578947	0.289474	−	0.957186i	$-0.406520\pi$
$19$	11.0000	0.578947	0.289474	+	0.957186i	$0.406520\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	39.5980i	1.72165i	0.508900	+	0.860826i	$0.330052\pi$
$23$	39.5980i	1.72165i	−0.508900	+	0.860826i	$0.669948\pi$
$24$	0	0
$25$	−7.00000	−0.280000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 33.9411i	− 1.17038i	−0.810895	−	0.585192i	$-0.801019\pi$
$29$	− 33.9411i	− 1.17038i	0.810895	−	0.585192i	$-0.198981\pi$
$30$	0	0
$31$	50.0000	1.61290	0.806452	−	0.591300i	$-0.201385\pi$
$31$	50.0000	1.61290	0.806452	+	0.591300i	$0.201385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.9706i	0.484873i
$36$	0	0
$37$	−33.0000	−0.891892	−0.445946	−	0.895060i	$-0.647133\pi$
$37$	−33.0000	−0.891892	−0.445946	+	0.895060i	$0.647133\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	33.9411i	0.827832i	0.910315	+	0.413916i	$0.135839\pi$
$41$	33.9411i	0.827832i	−0.910315	+	0.413916i	$0.864161\pi$
$42$	0	0
$43$	10.0000	0.232558	0.116279	−	0.993217i	$-0.462903\pi$
$43$	10.0000	0.232558	0.116279	+	0.993217i	$0.462903\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 84.8528i	− 1.80538i	−0.430293	−	0.902690i	$-0.641590\pi$
$47$	− 84.8528i	− 1.80538i	0.430293	−	0.902690i	$-0.358410\pi$
$48$	0	0
$49$	−40.0000	−0.816327
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.3137i	− 0.213466i	−0.994288	−	0.106733i	$-0.965961\pi$
$53$	− 11.3137i	− 0.213466i	0.994288	−	0.106733i	$-0.0340390\pi$
$54$	0	0
$55$	−32.0000	−0.581818
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 28.2843i	− 0.479394i	−0.970848	−	0.239697i	$-0.922952\pi$
$59$	− 28.2843i	− 0.479394i	0.970848	−	0.239697i	$-0.0770482\pi$
$60$	0	0
$61$	−41.0000	−0.672131	−0.336066	−	0.941839i	$-0.609096\pi$
$61$	−41.0000	−0.672131	−0.336066	+	0.941839i	$0.609096\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 96.1665i	− 1.47948i
$66$	0	0
$67$	83.0000	1.23881	0.619403	−	0.785073i	$-0.287375\pi$
$67$	83.0000	1.23881	0.619403	+	0.785073i	$0.287375\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 22.6274i	− 0.318696i	−0.987222	−	0.159348i	$-0.949061\pi$
$71$	− 22.6274i	− 0.318696i	0.987222	−	0.159348i	$-0.0509392\pi$
$72$	0	0
$73$	127.000	1.73973	0.869863	−	0.493293i	$-0.164207\pi$
$73$	127.000	1.73973	0.869863	+	0.493293i	$0.164207\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.9706i	0.220397i
$78$	0	0
$79$	19.0000	0.240506	0.120253	−	0.992743i	$-0.461629\pi$
$79$	19.0000	0.240506	0.120253	+	0.992743i	$0.461629\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 124.451i	− 1.49941i	−0.661774	−	0.749704i	$-0.730196\pi$
$83$	− 124.451i	− 1.49941i	0.661774	−	0.749704i	$-0.269804\pi$
$84$	0	0
$85$	−160.000	−1.88235
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 84.8528i	− 0.953402i	−0.879065	−	0.476701i	$-0.841832\pi$
$89$	− 84.8528i	− 0.953402i	0.879065	−	0.476701i	$-0.158168\pi$
$90$	0	0
$91$	−51.0000	−0.560440
$92$	0	0
$93$	0	0
$94$	0	0
$95$	62.2254i	0.655004i
$96$	0	0
$97$	167.000	1.72165	0.860825	−	0.508902i	$-0.169948\pi$
$97$	167.000	1.72165	0.860825	+	0.508902i	$0.169948\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−53.0000	−0.514563	−0.257282	−	0.966336i	$-0.582827\pi$
$103$	−53.0000	−0.514563	−0.257282	+	0.966336i	$0.582827\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	152.735i	1.42743i	0.700436	+	0.713715i	$0.252989\pi$
$107$	152.735i	1.42743i	−0.700436	+	0.713715i	$0.747011\pi$
$108$	0	0
$109$	10.0000	0.0917431	0.0458716	−	0.998947i	$-0.485394\pi$
$109$	10.0000	0.0917431	0.0458716	+	0.998947i	$0.485394\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 16.9706i	− 0.150182i	−0.997177	−	0.0750910i	$-0.976075\pi$
$113$	− 16.9706i	− 0.150182i	0.997177	−	0.0750910i	$-0.0239247\pi$
$114$	0	0
$115$	−224.000	−1.94783
$116$	0	0
$117$	0	0
$118$	0	0
$119$	84.8528i	0.713049i
$120$	0	0
$121$	89.0000	0.735537
$122$	0	0
$123$	0	0
$124$	0	0
$125$	101.823i	0.814587i
$126$	0	0
$127$	−110.000	−0.866142	−0.433071	−	0.901360i	$-0.642570\pi$
$127$	−110.000	−0.866142	−0.433071	+	0.901360i	$0.642570\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 56.5685i	− 0.431821i	−0.976413	−	0.215910i	$-0.930728\pi$
$131$	− 56.5685i	− 0.431821i	0.976413	−	0.215910i	$-0.0692719\pi$
$132$	0	0
$133$	33.0000	0.248120
$134$	0	0
$135$	0	0
$136$	0	0
$137$	152.735i	1.11485i	0.830226	+	0.557427i	$0.188211\pi$
$137$	152.735i	1.11485i	−0.830226	+	0.557427i	$0.811789\pi$
$138$	0	0
$139$	219.000	1.57554	0.787770	−	0.615970i	$-0.211236\pi$
$139$	219.000	1.57554	0.787770	+	0.615970i	$0.211236\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 96.1665i	− 0.672493i
$144$	0	0
$145$	192.000	1.32414
$146$	0	0
$147$	0	0
$148$	0	0
$149$	113.137i	0.759309i	0.925128	+	0.379655i	$0.123957\pi$
$149$	113.137i	0.759309i	−0.925128	+	0.379655i	$0.876043\pi$
$150$	0	0
$151$	11.0000	0.0728477	0.0364238	−	0.999336i	$-0.488403\pi$
$151$	11.0000	0.0728477	0.0364238	+	0.999336i	$0.488403\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	282.843i	1.82479i
$156$	0	0
$157$	−86.0000	−0.547771	−0.273885	−	0.961762i	$-0.588309\pi$
$157$	−86.0000	−0.547771	−0.273885	+	0.961762i	$0.588309\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	118.794i	0.737851i
$162$	0	0
$163$	−213.000	−1.30675	−0.653374	−	0.757035i	$-0.726647\pi$
$163$	−213.000	−1.30675	−0.653374	+	0.757035i	$0.726647\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	84.8528i	0.508101i	0.967191	+	0.254050i	$0.0817629\pi$
$167$	84.8528i	0.508101i	−0.967191	+	0.254050i	$0.918237\pi$
$168$	0	0
$169$	120.000	0.710059
$170$	0	0
$171$	0	0
$172$	0	0
$173$	169.706i	0.980957i	0.871453	+	0.490479i	$0.163178\pi$
$173$	169.706i	0.980957i	−0.871453	+	0.490479i	$0.836822\pi$
$174$	0	0
$175$	−21.0000	−0.120000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 203.647i	− 1.13769i	−0.822444	−	0.568846i	$-0.807390\pi$
$179$	− 203.647i	− 1.13769i	0.822444	−	0.568846i	$-0.192610\pi$
$180$	0	0
$181$	−9.00000	−0.0497238	−0.0248619	−	0.999691i	$-0.507915\pi$
$181$	−9.00000	−0.0497238	−0.0248619	+	0.999691i	$0.507915\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 186.676i	− 1.00906i
$186$	0	0
$187$	−160.000	−0.855615
$188$	0	0
$189$	0	0
$190$	0	0
$191$	84.8528i	0.444256i	0.975018	+	0.222128i	$0.0713002\pi$
$191$	84.8528i	0.444256i	−0.975018	+	0.222128i	$0.928700\pi$
$192$	0	0
$193$	47.0000	0.243523	0.121762	−	0.992559i	$-0.461146\pi$
$193$	47.0000	0.243523	0.121762	+	0.992559i	$0.461146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 84.8528i	− 0.430725i	−0.976534	−	0.215362i	$-0.930907\pi$
$197$	− 84.8528i	− 0.430725i	0.976534	−	0.215362i	$-0.0690933\pi$
$198$	0	0
$199$	−101.000	−0.507538	−0.253769	−	0.967265i	$-0.581670\pi$
$199$	−101.000	−0.507538	−0.253769	+	0.967265i	$0.581670\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 101.823i	− 0.501593i
$204$	0	0
$205$	−192.000	−0.936585
$206$	0	0
$207$	0	0
$208$	0	0
$209$	62.2254i	0.297729i
$210$	0	0
$211$	−29.0000	−0.137441	−0.0687204	−	0.997636i	$-0.521892\pi$
$211$	−29.0000	−0.137441	−0.0687204	+	0.997636i	$0.521892\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	56.5685i	0.263109i
$216$	0	0
$217$	150.000	0.691244
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 480.833i	− 2.17571i
$222$	0	0
$223$	370.000	1.65919	0.829596	−	0.558363i	$-0.188571\pi$
$223$	370.000	1.65919	0.829596	+	0.558363i	$0.188571\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	271.529i	1.19616i	0.801435	+	0.598082i	$0.204070\pi$
$227$	271.529i	1.19616i	−0.801435	+	0.598082i	$0.795930\pi$
$228$	0	0
$229$	−230.000	−1.00437	−0.502183	−	0.864761i	$-0.667470\pi$
$229$	−230.000	−1.00437	−0.502183	+	0.864761i	$0.667470\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 113.137i	− 0.485567i	−0.970081	−	0.242783i	$-0.921940\pi$
$233$	− 113.137i	− 0.485567i	0.970081	−	0.242783i	$-0.0780604\pi$
$234$	0	0
$235$	480.000	2.04255
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 429.921i	− 1.79883i	−0.437093	−	0.899416i	$-0.643992\pi$
$239$	− 429.921i	− 1.79883i	0.437093	−	0.899416i	$-0.356008\pi$
$240$	0	0
$241$	159.000	0.659751	0.329876	−	0.944024i	$-0.392993\pi$
$241$	159.000	0.659751	0.329876	+	0.944024i	$0.392993\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 226.274i	− 0.923568i
$246$	0	0
$247$	−187.000	−0.757085
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 260.215i	− 1.03671i	−0.855164	−	0.518357i	$-0.826544\pi$
$251$	− 260.215i	− 1.03671i	0.855164	−	0.518357i	$-0.173456\pi$
$252$	0	0
$253$	−224.000	−0.885375
$254$	0	0
$255$	0	0
$256$	0	0
$257$	282.843i	1.10056i	0.834982	+	0.550278i	$0.185478\pi$
$257$	282.843i	1.10056i	−0.834982	+	0.550278i	$0.814522\pi$
$258$	0	0
$259$	−99.0000	−0.382239
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 113.137i	− 0.430179i	−0.976594	−	0.215090i	$-0.930996\pi$
$263$	− 113.137i	− 0.430179i	0.976594	−	0.215090i	$-0.0690043\pi$
$264$	0	0
$265$	64.0000	0.241509
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 311.127i	− 1.15661i	−0.815822	−	0.578303i	$-0.803715\pi$
$269$	− 311.127i	− 1.15661i	0.815822	−	0.578303i	$-0.196285\pi$
$270$	0	0
$271$	59.0000	0.217712	0.108856	−	0.994058i	$-0.465281\pi$
$271$	59.0000	0.217712	0.108856	+	0.994058i	$0.465281\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 39.5980i	− 0.143993i
$276$	0	0
$277$	−230.000	−0.830325	−0.415162	−	0.909747i	$-0.636275\pi$
$277$	−230.000	−0.830325	−0.415162	+	0.909747i	$0.636275\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	282.843i	1.00656i	0.864124	+	0.503279i	$0.167873\pi$
$281$	282.843i	1.00656i	−0.864124	+	0.503279i	$0.832127\pi$
$282$	0	0
$283$	170.000	0.600707	0.300353	−	0.953828i	$-0.402895\pi$
$283$	170.000	0.600707	0.300353	+	0.953828i	$0.402895\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	101.823i	0.354785i
$288$	0	0
$289$	−511.000	−1.76817
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 16.9706i	− 0.0579200i	−0.999581	−	0.0289600i	$-0.990780\pi$
$293$	− 16.9706i	− 0.0579200i	0.999581	−	0.0289600i	$-0.00921954\pi$
$294$	0	0
$295$	160.000	0.542373
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 673.166i	− 2.25139i
$300$	0	0
$301$	30.0000	0.0996678
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 231.931i	− 0.760430i
$306$	0	0
$307$	186.000	0.605863	0.302932	−	0.953012i	$-0.402035\pi$
$307$	186.000	0.605863	0.302932	+	0.953012i	$0.402035\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 84.8528i	− 0.272839i	−0.990651	−	0.136419i	$-0.956441\pi$
$311$	− 84.8528i	− 0.272839i	0.990651	−	0.136419i	$-0.0435594\pi$
$312$	0	0
$313$	−393.000	−1.25559	−0.627796	−	0.778378i	$-0.716043\pi$
$313$	−393.000	−1.25559	−0.627796	+	0.778378i	$0.716043\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	509.117i	1.60605i	0.595947	+	0.803023i	$0.296777\pi$
$317$	509.117i	1.60605i	−0.595947	+	0.803023i	$0.703223\pi$
$318$	0	0
$319$	192.000	0.601881
$320$	0	0
$321$	0	0
$322$	0	0
$323$	311.127i	0.963241i
$324$	0	0
$325$	119.000	0.366154
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 254.558i	− 0.773734i
$330$	0	0
$331$	11.0000	0.0332326	0.0166163	−	0.999862i	$-0.494711\pi$
$331$	11.0000	0.0332326	0.0166163	+	0.999862i	$0.494711\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	469.519i	1.40155i
$336$	0	0
$337$	183.000	0.543027	0.271513	−	0.962435i	$-0.412476\pi$
$337$	183.000	0.543027	0.271513	+	0.962435i	$0.412476\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	282.843i	0.829451i
$342$	0	0
$343$	−267.000	−0.778426
$344$	0	0
$345$	0	0
$346$	0	0
$347$	610.940i	1.76063i	0.474385	+	0.880317i	$0.342670\pi$
$347$	610.940i	1.76063i	−0.474385	+	0.880317i	$0.657330\pi$
$348$	0	0
$349$	111.000	0.318052	0.159026	−	0.987274i	$-0.449165\pi$
$349$	111.000	0.318052	0.159026	+	0.987274i	$0.449165\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 565.685i	− 1.60251i	−0.598324	−	0.801254i	$-0.704166\pi$
$353$	− 565.685i	− 1.60251i	0.598324	−	0.801254i	$-0.295834\pi$
$354$	0	0
$355$	128.000	0.360563
$356$	0	0
$357$	0	0
$358$	0	0
$359$	118.794i	0.330902i	0.986218	+	0.165451i	$0.0529080\pi$
$359$	118.794i	0.330902i	−0.986218	+	0.165451i	$0.947092\pi$
$360$	0	0
$361$	−240.000	−0.664820
$362$	0	0
$363$	0	0
$364$	0	0
$365$	718.420i	1.96828i
$366$	0	0
$367$	667.000	1.81744	0.908719	−	0.417408i	$-0.137061\pi$
$367$	667.000	1.81744	0.908719	+	0.417408i	$0.137061\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 33.9411i	− 0.0914855i
$372$	0	0
$373$	623.000	1.67024	0.835121	−	0.550067i	$-0.185398\pi$
$373$	623.000	1.67024	0.835121	+	0.550067i	$0.185398\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	576.999i	1.53050i
$378$	0	0
$379$	371.000	0.978892	0.489446	−	0.872034i	$-0.337199\pi$
$379$	371.000	0.978892	0.489446	+	0.872034i	$0.337199\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	−96.0000	−0.249351
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 367.696i	− 0.945233i	−0.881268	−	0.472616i	$-0.843310\pi$
$389$	− 367.696i	− 0.945233i	0.881268	−	0.472616i	$-0.156690\pi$
$390$	0	0
$391$	−1120.00	−2.86445
$392$	0	0
$393$	0	0
$394$	0	0
$395$	107.480i	0.272102i
$396$	0	0
$397$	−310.000	−0.780856	−0.390428	−	0.920633i	$-0.627673\pi$
$397$	−310.000	−0.780856	−0.390428	+	0.920633i	$0.627673\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	135.765i	0.338565i	0.985568	+	0.169282i	$0.0541450\pi$
$401$	135.765i	0.338565i	−0.985568	+	0.169282i	$0.945855\pi$
$402$	0	0
$403$	−850.000	−2.10918
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 186.676i	− 0.458664i
$408$	0	0
$409$	−369.000	−0.902200	−0.451100	−	0.892473i	$-0.648968\pi$
$409$	−369.000	−0.902200	−0.451100	+	0.892473i	$0.648968\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 84.8528i	− 0.205455i
$414$	0	0
$415$	704.000	1.69639
$416$	0	0
$417$	0	0
$418$	0	0
$419$	763.675i	1.82261i	0.411727	+	0.911307i	$0.364926\pi$
$419$	763.675i	1.82261i	−0.411727	+	0.911307i	$0.635074\pi$
$420$	0	0
$421$	−49.0000	−0.116390	−0.0581948	−	0.998305i	$-0.518534\pi$
$421$	−49.0000	−0.116390	−0.0581948	+	0.998305i	$0.518534\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 197.990i	− 0.465859i
$426$	0	0
$427$	−123.000	−0.288056
$428$	0	0
$429$	0	0
$430$	0	0
$431$	503.460i	1.16812i	0.811710	+	0.584060i	$0.198537\pi$
$431$	503.460i	1.16812i	−0.811710	+	0.584060i	$0.801463\pi$
$432$	0	0
$433$	130.000	0.300231	0.150115	−	0.988668i	$-0.452035\pi$
$433$	130.000	0.300231	0.150115	+	0.988668i	$0.452035\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	435.578i	0.996745i
$438$	0	0
$439$	450.000	1.02506	0.512528	−	0.858670i	$-0.328709\pi$
$439$	450.000	1.02506	0.512528	+	0.858670i	$0.328709\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 56.5685i	− 0.127694i	−0.997960	−	0.0638471i	$-0.979663\pi$
$443$	− 56.5685i	− 0.127694i	0.997960	−	0.0638471i	$-0.0203370\pi$
$444$	0	0
$445$	480.000	1.07865
$446$	0	0
$447$	0	0
$448$	0	0
$449$	141.421i	0.314970i	0.987521	+	0.157485i	$0.0503385\pi$
$449$	141.421i	0.314970i	−0.987521	+	0.157485i	$0.949661\pi$
$450$	0	0
$451$	−192.000	−0.425721
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 288.500i	− 0.634065i
$456$	0	0
$457$	50.0000	0.109409	0.0547046	−	0.998503i	$-0.482578\pi$
$457$	50.0000	0.109409	0.0547046	+	0.998503i	$0.482578\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 367.696i	− 0.797604i	−0.917037	−	0.398802i	$-0.869426\pi$
$461$	− 367.696i	− 0.797604i	0.917037	−	0.398802i	$-0.130574\pi$
$462$	0	0
$463$	−637.000	−1.37581	−0.687905	−	0.725801i	$-0.741469\pi$
$463$	−637.000	−1.37581	−0.687905	+	0.725801i	$0.741469\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 593.970i	− 1.27188i	−0.771737	−	0.635942i	$-0.780612\pi$
$467$	− 593.970i	− 1.27188i	0.771737	−	0.635942i	$-0.219388\pi$
$468$	0	0
$469$	249.000	0.530917
$470$	0	0
$471$	0	0
$472$	0	0
$473$	56.5685i	0.119595i
$474$	0	0
$475$	−77.0000	−0.162105
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 599.627i	− 1.25183i	−0.779891	−	0.625915i	$-0.784726\pi$
$479$	− 599.627i	− 1.25183i	0.779891	−	0.625915i	$-0.215274\pi$
$480$	0	0
$481$	561.000	1.16632
$482$	0	0
$483$	0	0
$484$	0	0
$485$	944.695i	1.94782i
$486$	0	0
$487$	−557.000	−1.14374	−0.571869	−	0.820345i	$-0.693781\pi$
$487$	−557.000	−1.14374	−0.571869	+	0.820345i	$0.693781\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 560.029i	− 1.14059i	−0.821441	−	0.570294i	$-0.806829\pi$
$491$	− 560.029i	− 1.14059i	0.821441	−	0.570294i	$-0.193171\pi$
$492$	0	0
$493$	960.000	1.94726
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 67.8823i	− 0.136584i
$498$	0	0
$499$	−102.000	−0.204409	−0.102204	−	0.994763i	$-0.532590\pi$
$499$	−102.000	−0.204409	−0.102204	+	0.994763i	$0.532590\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 492.146i	− 0.978422i	−0.872165	−	0.489211i	$-0.837285\pi$
$503$	− 492.146i	− 0.978422i	0.872165	−	0.489211i	$-0.162715\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 729.734i	− 1.43366i	−0.697247	−	0.716831i	$-0.745592\pi$
$509$	− 729.734i	− 1.43366i	0.697247	−	0.716831i	$-0.254408\pi$
$510$	0	0
$511$	381.000	0.745597
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 299.813i	− 0.582162i
$516$	0	0
$517$	480.000	0.928433
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 684.479i	− 1.31378i	−0.753986	−	0.656890i	$-0.771872\pi$
$521$	− 684.479i	− 1.31378i	0.753986	−	0.656890i	$-0.228128\pi$
$522$	0	0
$523$	163.000	0.311663	0.155832	−	0.987784i	$-0.450194\pi$
$523$	163.000	0.311663	0.155832	+	0.987784i	$0.450194\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1414.21i	2.68352i
$528$	0	0
$529$	−1039.00	−1.96408
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 576.999i	− 1.08255i
$534$	0	0
$535$	−864.000	−1.61495
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 226.274i	− 0.419804i
$540$	0	0
$541$	351.000	0.648799	0.324399	−	0.945920i	$-0.394838\pi$
$541$	351.000	0.648799	0.324399	+	0.945920i	$0.394838\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	56.5685i	0.103795i
$546$	0	0
$547$	−997.000	−1.82267	−0.911335	−	0.411667i	$-0.864947\pi$
$547$	−997.000	−1.82267	−0.911335	+	0.411667i	$0.864947\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 373.352i	− 0.677591i
$552$	0	0
$553$	57.0000	0.103074
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 537.401i	− 0.964814i	−0.875947	−	0.482407i	$-0.839763\pi$
$557$	− 537.401i	− 0.964814i	0.875947	−	0.482407i	$-0.160237\pi$
$558$	0	0
$559$	−170.000	−0.304114
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 622.254i	− 1.10525i	−0.833431	−	0.552623i	$-0.813627\pi$
$563$	− 622.254i	− 1.10525i	0.833431	−	0.552623i	$-0.186373\pi$
$564$	0	0
$565$	96.0000	0.169912
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 367.696i	− 0.646214i	−0.946363	−	0.323107i	$-0.895273\pi$
$569$	− 367.696i	− 0.646214i	0.946363	−	0.323107i	$-0.104727\pi$
$570$	0	0
$571$	1131.00	1.98074	0.990368	−	0.138462i	$-0.0442158\pi$
$571$	1131.00	1.98074	0.990368	+	0.138462i	$0.0442158\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 277.186i	− 0.482062i
$576$	0	0
$577$	−937.000	−1.62392	−0.811958	−	0.583715i	$-0.801598\pi$
$577$	−937.000	−1.62392	−0.811958	+	0.583715i	$0.801598\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 373.352i	− 0.642603i
$582$	0	0
$583$	64.0000	0.109777
$584$	0	0
$585$	0	0
$586$	0	0
$587$	548.715i	0.934778i	0.884052	+	0.467389i	$0.154805\pi$
$587$	548.715i	0.934778i	−0.884052	+	0.467389i	$0.845195\pi$
$588$	0	0
$589$	550.000	0.933786
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 11.3137i	− 0.0190788i	−0.999954	−	0.00953938i	$-0.996963\pi$
$593$	− 11.3137i	− 0.0190788i	0.999954	−	0.00953938i	$-0.00303653\pi$
$594$	0	0
$595$	−480.000	−0.806723
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 644.881i	− 1.07660i	−0.842754	−	0.538298i	$-0.819067\pi$
$599$	− 644.881i	− 1.07660i	0.842754	−	0.538298i	$-0.180933\pi$
$600$	0	0
$601$	−110.000	−0.183028	−0.0915141	−	0.995804i	$-0.529171\pi$
$601$	−110.000	−0.183028	−0.0915141	+	0.995804i	$0.529171\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	503.460i	0.832165i
$606$	0	0
$607$	467.000	0.769357	0.384679	−	0.923051i	$-0.374312\pi$
$607$	467.000	0.769357	0.384679	+	0.923051i	$0.374312\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1442.50i	2.36088i
$612$	0	0
$613$	743.000	1.21207	0.606036	−	0.795437i	$-0.292759\pi$
$613$	743.000	1.21207	0.606036	+	0.795437i	$0.292759\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	526.087i	0.852654i	0.904569	+	0.426327i	$0.140193\pi$
$617$	526.087i	0.852654i	−0.904569	+	0.426327i	$0.859807\pi$
$618$	0	0
$619$	139.000	0.224556	0.112278	−	0.993677i	$-0.464185\pi$
$619$	139.000	0.224556	0.112278	+	0.993677i	$0.464185\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 254.558i	− 0.408601i
$624$	0	0
$625$	−751.000	−1.20160
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 933.381i	− 1.48391i
$630$	0	0
$631$	−101.000	−0.160063	−0.0800317	−	0.996792i	$-0.525502\pi$
$631$	−101.000	−0.160063	−0.0800317	+	0.996792i	$0.525502\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 622.254i	− 0.979928i
$636$	0	0
$637$	680.000	1.06750
$638$	0	0
$639$	0	0
$640$	0	0
$641$	565.685i	0.882505i	0.897383	+	0.441252i	$0.145466\pi$
$641$	565.685i	0.882505i	−0.897383	+	0.441252i	$0.854534\pi$
$642$	0	0
$643$	−870.000	−1.35303	−0.676516	−	0.736428i	$-0.736511\pi$
$643$	−870.000	−1.35303	−0.676516	+	0.736428i	$0.736511\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	350.725i	0.542079i	0.962568	+	0.271039i	$0.0873674\pi$
$647$	350.725i	0.542079i	−0.962568	+	0.271039i	$0.912633\pi$
$648$	0	0
$649$	160.000	0.246533
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 169.706i	− 0.259886i	−0.991521	−	0.129943i	$-0.958521\pi$
$653$	− 169.706i	− 0.259886i	0.991521	−	0.129943i	$-0.0414795\pi$
$654$	0	0
$655$	320.000	0.488550
$656$	0	0
$657$	0	0
$658$	0	0
$659$	169.706i	0.257520i	0.991676	+	0.128760i	$0.0410997\pi$
$659$	169.706i	0.257520i	−0.991676	+	0.128760i	$0.958900\pi$
$660$	0	0
$661$	−121.000	−0.183056	−0.0915280	−	0.995803i	$-0.529175\pi$
$661$	−121.000	−0.183056	−0.0915280	+	0.995803i	$0.529175\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	186.676i	0.280716i
$666$	0	0
$667$	1344.00	2.01499
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 231.931i	− 0.345650i
$672$	0	0
$673$	727.000	1.08024	0.540119	−	0.841589i	$-0.318379\pi$
$673$	727.000	1.08024	0.540119	+	0.841589i	$0.318379\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	311.127i	0.459567i	0.973242	+	0.229784i	$0.0738018\pi$
$677$	311.127i	0.459567i	−0.973242	+	0.229784i	$0.926198\pi$
$678$	0	0
$679$	501.000	0.737850
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 384.666i	− 0.563201i	−0.959532	−	0.281600i	$-0.909135\pi$
$683$	− 384.666i	− 0.563201i	0.959532	−	0.281600i	$-0.0908652\pi$
$684$	0	0
$685$	−864.000	−1.26131
$686$	0	0
$687$	0	0
$688$	0	0
$689$	192.333i	0.279148i
$690$	0	0
$691$	−70.0000	−0.101302	−0.0506512	−	0.998716i	$-0.516130\pi$
$691$	−70.0000	−0.101302	−0.0506512	+	0.998716i	$0.516130\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1238.85i	1.78252i
$696$	0	0
$697$	−960.000	−1.37733
$698$	0	0
$699$	0	0
$700$	0	0
$701$	729.734i	1.04099i	0.853865	+	0.520495i	$0.174253\pi$
$701$	729.734i	1.04099i	−0.853865	+	0.520495i	$0.825747\pi$
$702$	0	0
$703$	−363.000	−0.516358
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−961.000	−1.35543	−0.677715	−	0.735325i	$-0.737030\pi$
$709$	−961.000	−1.35543	−0.677715	+	0.735325i	$0.737030\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1979.90i	2.77686i
$714$	0	0
$715$	544.000	0.760839
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 758.018i	− 1.05427i	−0.849782	−	0.527134i	$-0.823267\pi$
$719$	− 758.018i	− 1.05427i	0.849782	−	0.527134i	$-0.176733\pi$
$720$	0	0
$721$	−159.000	−0.220527
$722$	0	0
$723$	0	0
$724$	0	0
$725$	237.588i	0.327707i
$726$	0	0
$727$	−510.000	−0.701513	−0.350757	−	0.936467i	$-0.614076\pi$
$727$	−510.000	−0.701513	−0.350757	+	0.936467i	$0.614076\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	282.843i	0.386926i
$732$	0	0
$733$	−790.000	−1.07776	−0.538881	−	0.842382i	$-0.681153\pi$
$733$	−790.000	−1.07776	−0.538881	+	0.842382i	$0.681153\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	469.519i	0.637068i
$738$	0	0
$739$	410.000	0.554804	0.277402	−	0.960754i	$-0.410527\pi$
$739$	410.000	0.554804	0.277402	+	0.960754i	$0.410527\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1159.66i	1.56077i	0.625297	+	0.780387i	$0.284978\pi$
$743$	1159.66i	1.56077i	−0.625297	+	0.780387i	$0.715022\pi$
$744$	0	0
$745$	−640.000	−0.859060
$746$	0	0
$747$	0	0
$748$	0	0
$749$	458.205i	0.611756i
$750$	0	0
$751$	99.0000	0.131824	0.0659121	−	0.997825i	$-0.479004\pi$
$751$	99.0000	0.131824	0.0659121	+	0.997825i	$0.479004\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	62.2254i	0.0824177i
$756$	0	0
$757$	1007.00	1.33025	0.665125	−	0.746732i	$-0.268378\pi$
$757$	1007.00	1.33025	0.665125	+	0.746732i	$0.268378\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1193.60i	− 1.56846i	−0.620472	−	0.784229i	$-0.713059\pi$
$761$	− 1193.60i	− 1.56846i	0.620472	−	0.784229i	$-0.286941\pi$
$762$	0	0
$763$	30.0000	0.0393185
$764$	0	0
$765$	0	0
$766$	0	0
$767$	480.833i	0.626900i
$768$	0	0
$769$	−481.000	−0.625488	−0.312744	−	0.949838i	$-0.601248\pi$
$769$	−481.000	−0.625488	−0.312744	+	0.949838i	$0.601248\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 780.646i	− 1.00989i	−0.863151	−	0.504946i	$-0.831513\pi$
$773$	− 780.646i	− 1.00989i	0.863151	−	0.504946i	$-0.168487\pi$
$774$	0	0
$775$	−350.000	−0.451613
$776$	0	0
$777$	0	0
$778$	0	0
$779$	373.352i	0.479271i
$780$	0	0
$781$	128.000	0.163892
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 486.489i	− 0.619732i
$786$	0	0
$787$	667.000	0.847522	0.423761	−	0.905774i	$-0.360710\pi$
$787$	667.000	0.847522	0.423761	+	0.905774i	$0.360710\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 50.9117i	− 0.0643637i
$792$	0	0
$793$	697.000	0.878941
$794$	0	0
$795$	0	0
$796$	0	0
$797$	565.685i	0.709768i	0.934910	+	0.354884i	$0.115480\pi$
$797$	565.685i	0.709768i	−0.934910	+	0.354884i	$0.884520\pi$
$798$	0	0
$799$	2400.00	3.00375
$800$	0	0
$801$	0	0
$802$	0	0
$803$	718.420i	0.894671i
$804$	0	0
$805$	−672.000	−0.834783
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 565.685i	− 0.699240i	−0.936892	−	0.349620i	$-0.886311\pi$
$809$	− 565.685i	− 0.699240i	0.936892	−	0.349620i	$-0.113689\pi$
$810$	0	0
$811$	−950.000	−1.17139	−0.585697	−	0.810530i	$-0.699179\pi$
$811$	−950.000	−1.17139	−0.585697	+	0.810530i	$0.699179\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1204.91i	− 1.47842i
$816$	0	0
$817$	110.000	0.134639
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1272.79i	1.55030i	0.631780	+	0.775148i	$0.282324\pi$
$821$	1272.79i	1.55030i	−0.631780	+	0.775148i	$0.717676\pi$
$822$	0	0
$823$	923.000	1.12151	0.560753	−	0.827983i	$-0.310512\pi$
$823$	923.000	1.12151	0.560753	+	0.827983i	$0.310512\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	876.812i	1.06023i	0.847925	+	0.530116i	$0.177852\pi$
$827$	876.812i	1.06023i	−0.847925	+	0.530116i	$0.822148\pi$
$828$	0	0
$829$	−281.000	−0.338963	−0.169481	−	0.985533i	$-0.554209\pi$
$829$	−281.000	−0.338963	−0.169481	+	0.985533i	$0.554209\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1131.37i	− 1.35819i
$834$	0	0
$835$	−480.000	−0.574850
$836$	0	0
$837$	0	0
$838$	0	0
$839$	509.117i	0.606814i	0.952861	+	0.303407i	$0.0981242\pi$
$839$	509.117i	0.606814i	−0.952861	+	0.303407i	$0.901876\pi$
$840$	0	0
$841$	−311.000	−0.369798
$842$	0	0
$843$	0	0
$844$	0	0
$845$	678.823i	0.803340i
$846$	0	0
$847$	267.000	0.315230
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1306.73i	− 1.53553i
$852$	0	0
$853$	−337.000	−0.395076	−0.197538	−	0.980295i	$-0.563295\pi$
$853$	−337.000	−0.395076	−0.197538	+	0.980295i	$0.563295\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1086.12i	− 1.26735i	−0.773601	−	0.633673i	$-0.781546\pi$
$857$	− 1086.12i	− 1.26735i	0.773601	−	0.633673i	$-0.218454\pi$
$858$	0	0
$859$	1299.00	1.51222	0.756112	−	0.654443i	$-0.227097\pi$
$859$	1299.00	1.51222	0.756112	+	0.654443i	$0.227097\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 197.990i	− 0.229421i	−0.993399	−	0.114710i	$-0.963406\pi$
$863$	− 197.990i	− 0.229421i	0.993399	−	0.114710i	$-0.0365940\pi$
$864$	0	0
$865$	−960.000	−1.10983
$866$	0	0
$867$	0	0
$868$	0	0
$869$	107.480i	0.123683i
$870$	0	0
$871$	−1411.00	−1.61998
$872$	0	0
$873$	0	0
$874$	0	0
$875$	305.470i	0.349109i
$876$	0	0
$877$	303.000	0.345496	0.172748	−	0.984966i	$-0.444735\pi$
$877$	303.000	0.345496	0.172748	+	0.984966i	$0.444735\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 5.65685i	− 0.00642095i	−0.999995	−	0.00321047i	$-0.998978\pi$
$881$	− 5.65685i	− 0.00642095i	0.999995	−	0.00321047i	$-0.00102193\pi$
$882$	0	0
$883$	−613.000	−0.694224	−0.347112	−	0.937824i	$-0.612838\pi$
$883$	−613.000	−0.694224	−0.347112	+	0.937824i	$0.612838\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	282.843i	0.318876i	0.987208	+	0.159438i	$0.0509682\pi$
$887$	282.843i	0.318876i	−0.987208	+	0.159438i	$0.949032\pi$
$888$	0	0
$889$	−330.000	−0.371204
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 933.381i	− 1.04522i
$894$	0	0
$895$	1152.00	1.28715
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1697.06i	− 1.88772i
$900$	0	0
$901$	320.000	0.355161
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 50.9117i	− 0.0562560i
$906$	0	0
$907$	−173.000	−0.190739	−0.0953693	−	0.995442i	$-0.530403\pi$
$907$	−173.000	−0.190739	−0.0953693	+	0.995442i	$0.530403\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 282.843i	− 0.310475i	−0.987877	−	0.155237i	$-0.950386\pi$
$911$	− 282.843i	− 0.310475i	0.987877	−	0.155237i	$-0.0496143\pi$
$912$	0	0
$913$	704.000	0.771084
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 169.706i	− 0.185066i
$918$	0	0
$919$	−830.000	−0.903156	−0.451578	−	0.892232i	$-0.649139\pi$
$919$	−830.000	−0.903156	−0.451578	+	0.892232i	$0.649139\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	384.666i	0.416756i
$924$	0	0
$925$	231.000	0.249730
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 769.332i	− 0.828129i	−0.910247	−	0.414065i	$-0.864109\pi$
$929$	− 769.332i	− 0.828129i	0.910247	−	0.414065i	$-0.135891\pi$
$930$	0	0
$931$	−440.000	−0.472610
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 905.097i	− 0.968018i
$936$	0	0
$937$	663.000	0.707577	0.353789	−	0.935325i	$-0.384893\pi$
$937$	663.000	0.707577	0.353789	+	0.935325i	$0.384893\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1069.15i	− 1.13618i	−0.822966	−	0.568090i	$-0.807683\pi$
$941$	− 1069.15i	− 1.13618i	0.822966	−	0.568090i	$-0.192317\pi$
$942$	0	0
$943$	−1344.00	−1.42524
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 933.381i	− 0.985619i	−0.870137	−	0.492809i	$-0.835970\pi$
$947$	− 933.381i	− 0.985619i	0.870137	−	0.492809i	$-0.164030\pi$
$948$	0	0
$949$	−2159.00	−2.27503
$950$	0	0
$951$	0	0
$952$	0	0
$953$	254.558i	0.267113i	0.991041	+	0.133556i	$0.0426397\pi$
$953$	254.558i	0.267113i	−0.991041	+	0.133556i	$0.957360\pi$
$954$	0	0
$955$	−480.000	−0.502618
$956$	0	0
$957$	0	0
$958$	0	0
$959$	458.205i	0.477795i
$960$	0	0
$961$	1539.00	1.60146
$962$	0	0
$963$	0	0
$964$	0	0
$965$	265.872i	0.275515i
$966$	0	0
$967$	−1357.00	−1.40331	−0.701655	−	0.712517i	$-0.747555\pi$
$967$	−1357.00	−1.40331	−0.701655	+	0.712517i	$0.747555\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 424.264i	− 0.436935i	−0.975844	−	0.218468i	$-0.929894\pi$
$971$	− 424.264i	− 0.436935i	0.975844	−	0.218468i	$-0.0701058\pi$
$972$	0	0
$973$	657.000	0.675231
$974$	0	0
$975$	0	0
$976$	0	0
$977$	916.410i	0.937984i	0.883202	+	0.468992i	$0.155383\pi$
$977$	916.410i	0.937984i	−0.883202	+	0.468992i	$0.844617\pi$
$978$	0	0
$979$	480.000	0.490296
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1204.91i	− 1.22575i	−0.790181	−	0.612874i	$-0.790013\pi$
$983$	− 1204.91i	− 1.22575i	0.790181	−	0.612874i	$-0.209987\pi$
$984$	0	0
$985$	480.000	0.487310
$986$	0	0
$987$	0	0
$988$	0	0
$989$	395.980i	0.400384i
$990$	0	0
$991$	1531.00	1.54490	0.772452	−	0.635073i	$-0.219030\pi$
$991$	1531.00	1.54490	0.772452	+	0.635073i	$0.219030\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 571.342i	− 0.574213i
$996$	0	0
$997$	1210.00	1.21364	0.606820	−	0.794839i	$-0.292445\pi$
$997$	1210.00	1.21364	0.606820	+	0.794839i	$0.292445\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.e.b.161.2	yes	2
3.2	odd	2	inner	216.3.e.b.161.1	✓	2
4.3	odd	2		432.3.e.e.161.2		2
8.3	odd	2		1728.3.e.i.1025.1		2
8.5	even	2		1728.3.e.k.1025.1		2
9.2	odd	6		648.3.m.b.377.2		4
9.4	even	3		648.3.m.b.593.2		4
9.5	odd	6		648.3.m.b.593.1		4
9.7	even	3		648.3.m.b.377.1		4
12.11	even	2		432.3.e.e.161.1		2
24.5	odd	2		1728.3.e.k.1025.2		2
24.11	even	2		1728.3.e.i.1025.2		2
36.7	odd	6		1296.3.q.h.1025.1		4
36.11	even	6		1296.3.q.h.1025.2		4
36.23	even	6		1296.3.q.h.593.1		4
36.31	odd	6		1296.3.q.h.593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.e.b.161.1	✓	2	3.2	odd	2	inner
216.3.e.b.161.2	yes	2	1.1	even	1	trivial
432.3.e.e.161.1		2	12.11	even	2
432.3.e.e.161.2		2	4.3	odd	2
648.3.m.b.377.1		4	9.7	even	3
648.3.m.b.377.2		4	9.2	odd	6
648.3.m.b.593.1		4	9.5	odd	6
648.3.m.b.593.2		4	9.4	even	3
1296.3.q.h.593.1		4	36.23	even	6
1296.3.q.h.593.2		4	36.31	odd	6
1296.3.q.h.1025.1		4	36.7	odd	6
1296.3.q.h.1025.2		4	36.11	even	6
1728.3.e.i.1025.1		2	8.3	odd	2
1728.3.e.i.1025.2		2	24.11	even	2
1728.3.e.k.1025.1		2	8.5	even	2
1728.3.e.k.1025.2		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.65685i q^{5} +3.00000 q^{7} +O(q^{10})\)	\(q+5.65685i q^{5} +3.00000 q^{7} +5.65685i q^{11} -17.0000 q^{13} +28.2843i q^{17} +11.0000 q^{19} +39.5980i q^{23} -7.00000 q^{25} -33.9411i q^{29} +50.0000 q^{31} +16.9706i q^{35} -33.0000 q^{37} +33.9411i q^{41} +10.0000 q^{43} -84.8528i q^{47} -40.0000 q^{49} -11.3137i q^{53} -32.0000 q^{55} -28.2843i q^{59} -41.0000 q^{61} -96.1665i q^{65} +83.0000 q^{67} -22.6274i q^{71} +127.000 q^{73} +16.9706i q^{77} +19.0000 q^{79} -124.451i q^{83} -160.000 q^{85} -84.8528i q^{89} -51.0000 q^{91} +62.2254i q^{95} +167.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{7}+O(q^{10}) \) 2 * q + 6 * q^7	\( 2 q + 6 q^{7} - 34 q^{13} + 22 q^{19} - 14 q^{25} + 100 q^{31} - 66 q^{37} + 20 q^{43} - 80 q^{49} - 64 q^{55} - 82 q^{61} + 166 q^{67} + 254 q^{73} + 38 q^{79} - 320 q^{85} - 102 q^{91} + 334 q^{97}+O(q^{100}) \) 2 * q + 6 * q^7 - 34 * q^13 + 22 * q^19 - 14 * q^25 + 100 * q^31 - 66 * q^37 + 20 * q^43 - 80 * q^49 - 64 * q^55 - 82 * q^61 + 166 * q^67 + 254 * q^73 + 38 * q^79 - 320 * q^85 - 102 * q^91 + 334 * q^97

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