LMFDB - Embedded newform 8208.2.a.bg.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,1,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$65.5412099791$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2052)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05408 q^{5} +2.46050 q^{7} -1.46050 q^{11} -3.64766 q^{13} -5.92101 q^{17} -1.00000 q^{19} +0.320233 q^{23} -0.780738 q^{25} -0.945916 q^{29} +7.19436 q^{31} +5.05408 q^{35} +2.59358 q^{37} -6.86693 q^{41} -5.78794 q^{43} +1.81284 q^{47} -0.945916 q^{49} -5.96790 q^{53} -3.00000 q^{55} +11.2484 q^{59} +2.83482 q^{61} -7.49261 q^{65} -6.62276 q^{67} -2.64766 q^{71} +3.81284 q^{73} -3.59358 q^{77} -7.24844 q^{79} +0.0789903 q^{83} -12.1623 q^{85} -12.7558 q^{89} -8.97509 q^{91} -2.05408 q^{95} -10.9823 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + q^{7} + 2 q^{11} + q^{13} - 5 q^{17} - 3 q^{19} - q^{23} + 6 q^{25} - 12 q^{29} + 8 q^{31} + 6 q^{35} + 5 q^{37} - 17 q^{41} - q^{43} + 11 q^{47} - 12 q^{49} - 4 q^{53} - 9 q^{55} + 11 q^{59}+ \cdots - 3 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + q^7 + 2 * q^11 + q^13 - 5 * q^17 - 3 * q^19 - q^23 + 6 * q^25 - 12 * q^29 + 8 * q^31 + 6 * q^35 + 5 * q^37 - 17 * q^41 - q^43 + 11 * q^47 - 12 * q^49 - 4 * q^53 - 9 * q^55 + 11 * q^59 - 9 * q^61 - 30 * q^65 + 14 * q^67 + 4 * q^71 + 17 * q^73 - 8 * q^77 + q^79 + 13 * q^83 - 9 * q^85 - 8 * q^89 - 5 * q^91 + 3 * q^95 - 3 * q^97

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
$5$	2.05408	0.918614	0.459307	−	0.888277i	$-0.348098\pi$
$5$	2.05408	0.918614	0.459307	+	0.888277i	$0.348098\pi$
$6$	0	0
$7$	2.46050	0.929983	0.464992	−	0.885315i	$-0.346057\pi$
$7$	2.46050	0.929983	0.464992	+	0.885315i	$0.346057\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.46050	−0.440359	−0.220179	−	0.975459i	$-0.570664\pi$
$11$	−1.46050	−0.440359	−0.220179	+	0.975459i	$0.570664\pi$
$12$	0	0
$13$	−3.64766	−1.01168	−0.505840	−	0.862627i	$-0.668817\pi$
$13$	−3.64766	−1.01168	−0.505840	+	0.862627i	$0.668817\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.92101	−1.43606	−0.718028	−	0.696014i	$-0.754955\pi$
$17$	−5.92101	−1.43606	−0.718028	+	0.696014i	$0.754955\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.320233	0.0667732	0.0333866	−	0.999443i	$-0.489371\pi$
$23$	0.320233	0.0667732	0.0333866	+	0.999443i	$0.489371\pi$
$24$	0	0
$25$	−0.780738	−0.156148
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.945916	−0.175652	−0.0878261	−	0.996136i	$-0.527992\pi$
$29$	−0.945916	−0.175652	−0.0878261	+	0.996136i	$0.527992\pi$
$30$	0	0
$31$	7.19436	1.29214	0.646072	−	0.763276i	$-0.276410\pi$
$31$	7.19436	1.29214	0.646072	+	0.763276i	$0.276410\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.05408	0.854296
$36$	0	0
$37$	2.59358	0.426382	0.213191	−	0.977011i	$-0.431614\pi$
$37$	2.59358	0.426382	0.213191	+	0.977011i	$0.431614\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.86693	−1.07243	−0.536217	−	0.844080i	$-0.680147\pi$
$41$	−6.86693	−1.07243	−0.536217	+	0.844080i	$0.680147\pi$
$42$	0	0
$43$	−5.78794	−0.882652	−0.441326	−	0.897347i	$-0.645492\pi$
$43$	−5.78794	−0.882652	−0.441326	+	0.897347i	$0.645492\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.81284	0.264430	0.132215	−	0.991221i	$-0.457791\pi$
$47$	1.81284	0.264430	0.132215	+	0.991221i	$0.457791\pi$
$48$	0	0
$49$	−0.945916	−0.135131
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.96790	−0.819754	−0.409877	−	0.912141i	$-0.634428\pi$
$53$	−5.96790	−0.819754	−0.409877	+	0.912141i	$0.634428\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.2484	1.46442	0.732211	−	0.681078i	$-0.238488\pi$
$59$	11.2484	1.46442	0.732211	+	0.681078i	$0.238488\pi$
$60$	0	0
$61$	2.83482	0.362962	0.181481	−	0.983394i	$-0.441911\pi$
$61$	2.83482	0.362962	0.181481	+	0.983394i	$0.441911\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.49261	−0.929344
$66$	0	0
$67$	−6.62276	−0.809099	−0.404549	−	0.914516i	$-0.632572\pi$
$67$	−6.62276	−0.809099	−0.404549	+	0.914516i	$0.632572\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.64766	−0.314220	−0.157110	−	0.987581i	$-0.550218\pi$
$71$	−2.64766	−0.314220	−0.157110	+	0.987581i	$0.550218\pi$
$72$	0	0
$73$	3.81284	0.446259	0.223130	−	0.974789i	$-0.428373\pi$
$73$	3.81284	0.446259	0.223130	+	0.974789i	$0.428373\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.59358	−0.409526
$78$	0	0
$79$	−7.24844	−0.815513	−0.407757	−	0.913091i	$-0.633689\pi$
$79$	−7.24844	−0.815513	−0.407757	+	0.913091i	$0.633689\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.0789903	0.00867031	0.00433515	−	0.999991i	$-0.498620\pi$
$83$	0.0789903	0.00867031	0.00433515	+	0.999991i	$0.498620\pi$
$84$	0	0
$85$	−12.1623	−1.31918
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.7558	−1.35212	−0.676058	−	0.736849i	$-0.736313\pi$
$89$	−12.7558	−1.35212	−0.676058	+	0.736849i	$0.736313\pi$
$90$	0	0
$91$	−8.97509	−0.940845
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.05408	−0.210745
$96$	0	0
$97$	−10.9823	−1.11508	−0.557541	−	0.830149i	$-0.688255\pi$
$97$	−10.9823	−1.11508	−0.557541	+	0.830149i	$0.688255\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.194356	0.0193391	0.00966957	−	0.999953i	$-0.496922\pi$
$101$	0.194356	0.0193391	0.00966957	+	0.999953i	$0.496922\pi$
$102$	0	0
$103$	1.32023	0.130086	0.0650432	−	0.997882i	$-0.479281\pi$
$103$	1.32023	0.130086	0.0650432	+	0.997882i	$0.479281\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0833	−1.16813	−0.584067	−	0.811706i	$-0.698539\pi$
$107$	−12.0833	−1.16813	−0.584067	+	0.811706i	$0.698539\pi$
$108$	0	0
$109$	14.1623	1.35650	0.678249	−	0.734832i	$-0.262739\pi$
$109$	14.1623	1.35650	0.678249	+	0.734832i	$0.262739\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.59358	−0.902488	−0.451244	−	0.892401i	$-0.649020\pi$
$113$	−9.59358	−0.902488	−0.451244	+	0.892401i	$0.649020\pi$
$114$	0	0
$115$	0.657786	0.0613389
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−14.5687	−1.33551
$120$	0	0
$121$	−8.86693	−0.806084
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.8741	−1.06205
$126$	0	0
$127$	15.0584	1.33621	0.668107	−	0.744066i	$-0.267105\pi$
$127$	15.0584	1.33621	0.668107	+	0.744066i	$0.267105\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.43560	0.824392	0.412196	−	0.911095i	$-0.364762\pi$
$131$	9.43560	0.824392	0.412196	+	0.911095i	$0.364762\pi$
$132$	0	0
$133$	−2.46050	−0.213353
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.8348	−1.09655	−0.548276	−	0.836297i	$-0.684716\pi$
$137$	−12.8348	−1.09655	−0.548276	+	0.836297i	$0.684716\pi$
$138$	0	0
$139$	−7.21634	−0.612082	−0.306041	−	0.952018i	$-0.599004\pi$
$139$	−7.21634	−0.612082	−0.306041	+	0.952018i	$0.599004\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.32743	0.445502
$144$	0	0
$145$	−1.94299	−0.161357
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.75583	0.307690	0.153845	−	0.988095i	$-0.450834\pi$
$149$	3.75583	0.307690	0.153845	+	0.988095i	$0.450834\pi$
$150$	0	0
$151$	−7.24844	−0.589870	−0.294935	−	0.955517i	$-0.595298\pi$
$151$	−7.24844	−0.589870	−0.294935	+	0.955517i	$0.595298\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	14.7778	1.18698
$156$	0	0
$157$	−9.37432	−0.748152	−0.374076	−	0.927398i	$-0.622040\pi$
$157$	−9.37432	−0.748152	−0.374076	+	0.927398i	$0.622040\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.787935	0.0620980
$162$	0	0
$163$	−21.2704	−1.66603	−0.833014	−	0.553252i	$-0.813387\pi$
$163$	−21.2704	−1.66603	−0.833014	+	0.553252i	$0.813387\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.21926	−0.326496	−0.163248	−	0.986585i	$-0.552197\pi$
$167$	−4.21926	−0.326496	−0.163248	+	0.986585i	$0.552197\pi$
$168$	0	0
$169$	0.305449	0.0234961
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.67257	−0.279220	−0.139610	−	0.990207i	$-0.544585\pi$
$173$	−3.67257	−0.279220	−0.139610	+	0.990207i	$0.544585\pi$
$174$	0	0
$175$	−1.92101	−0.145215
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.7702	−1.25347	−0.626733	−	0.779234i	$-0.715608\pi$
$179$	−16.7702	−1.25347	−0.626733	+	0.779234i	$0.715608\pi$
$180$	0	0
$181$	20.6768	1.53690	0.768449	−	0.639911i	$-0.221029\pi$
$181$	20.6768	1.53690	0.768449	+	0.639911i	$0.221029\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.32743	0.391680
$186$	0	0
$187$	8.64766	0.632380
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.0220	−1.23167	−0.615834	−	0.787876i	$-0.711181\pi$
$191$	−17.0220	−1.23167	−0.615834	+	0.787876i	$0.711181\pi$
$192$	0	0
$193$	−7.27335	−0.523547	−0.261773	−	0.965129i	$-0.584307\pi$
$193$	−7.27335	−0.523547	−0.261773	+	0.965129i	$0.584307\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.7381	−1.83377	−0.916883	−	0.399157i	$-0.869303\pi$
$197$	−25.7381	−1.83377	−0.916883	+	0.399157i	$0.869303\pi$
$198$	0	0
$199$	15.9253	1.12891	0.564457	−	0.825463i	$-0.309086\pi$
$199$	15.9253	1.12891	0.564457	+	0.825463i	$0.309086\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.32743	−0.163354
$204$	0	0
$205$	−14.1052	−0.985153
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.46050	0.101025
$210$	0	0
$211$	8.07607	0.555979	0.277990	−	0.960584i	$-0.410332\pi$
$211$	8.07607	0.555979	0.277990	+	0.960584i	$0.410332\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.8889	−0.810817
$216$	0	0
$217$	17.7017	1.20167
$218$	0	0
$219$	0	0
$220$	0	0
$221$	21.5979	1.45283
$222$	0	0
$223$	19.3566	1.29621	0.648107	−	0.761549i	$-0.275561\pi$
$223$	19.3566	1.29621	0.648107	+	0.761549i	$0.275561\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.7161	1.17586	0.587931	−	0.808911i	$-0.299943\pi$
$227$	17.7161	1.17586	0.587931	+	0.808911i	$0.299943\pi$
$228$	0	0
$229$	9.73812	0.643513	0.321757	−	0.946822i	$-0.395727\pi$
$229$	9.73812	0.643513	0.321757	+	0.946822i	$0.395727\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.83482	0.644301	0.322150	−	0.946689i	$-0.395594\pi$
$233$	9.83482	0.644301	0.322150	+	0.946689i	$0.395594\pi$
$234$	0	0
$235$	3.72373	0.242909
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.67257	−0.625667	−0.312833	−	0.949808i	$-0.601278\pi$
$239$	−9.67257	−0.625667	−0.312833	+	0.949808i	$0.601278\pi$
$240$	0	0
$241$	−13.0043	−0.837679	−0.418839	−	0.908060i	$-0.637563\pi$
$241$	−13.0043	−0.837679	−0.418839	+	0.908060i	$0.637563\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.94299	−0.124133
$246$	0	0
$247$	3.64766	0.232095
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.9282	0.879141	0.439570	−	0.898208i	$-0.355131\pi$
$251$	13.9282	0.879141	0.439570	+	0.898208i	$0.355131\pi$
$252$	0	0
$253$	−0.467702	−0.0294042
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.3274	−0.893721	−0.446860	−	0.894604i	$-0.647458\pi$
$257$	−14.3274	−0.893721	−0.446860	+	0.894604i	$0.647458\pi$
$258$	0	0
$259$	6.38151	0.396528
$260$	0	0
$261$	0	0
$262$	0	0
$263$	25.5801	1.57734	0.788670	−	0.614817i	$-0.210770\pi$
$263$	25.5801	1.57734	0.788670	+	0.614817i	$0.210770\pi$
$264$	0	0
$265$	−12.2586	−0.753038
$266$	0	0
$267$	0	0
$268$	0	0
$269$	25.6270	1.56251	0.781254	−	0.624214i	$-0.214580\pi$
$269$	25.6270	1.56251	0.781254	+	0.624214i	$0.214580\pi$
$270$	0	0
$271$	−17.5615	−1.06678	−0.533392	−	0.845868i	$-0.679083\pi$
$271$	−17.5615	−1.06678	−0.533392	+	0.845868i	$0.679083\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.14027	0.0687610
$276$	0	0
$277$	4.80992	0.289000	0.144500	−	0.989505i	$-0.453843\pi$
$277$	4.80992	0.289000	0.144500	+	0.989505i	$0.453843\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.4035	0.739931	0.369965	−	0.929046i	$-0.379370\pi$
$281$	12.4035	0.739931	0.369965	+	0.929046i	$0.379370\pi$
$282$	0	0
$283$	20.4969	1.21841	0.609207	−	0.793012i	$-0.291488\pi$
$283$	20.4969	1.21841	0.609207	+	0.793012i	$0.291488\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−16.8961	−0.997345
$288$	0	0
$289$	18.0584	1.06226
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−14.9679	−0.874434	−0.437217	−	0.899356i	$-0.644036\pi$
$293$	−14.9679	−0.874434	−0.437217	+	0.899356i	$0.644036\pi$
$294$	0	0
$295$	23.1052	1.34524
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.16810	−0.0675531
$300$	0	0
$301$	−14.2412	−0.820852
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.82296	0.333422
$306$	0	0
$307$	−4.29533	−0.245147	−0.122574	−	0.992459i	$-0.539115\pi$
$307$	−4.29533	−0.245147	−0.122574	+	0.992459i	$0.539115\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.67977	−0.492184	−0.246092	−	0.969246i	$-0.579147\pi$
$311$	−8.67977	−0.492184	−0.246092	+	0.969246i	$0.579147\pi$
$312$	0	0
$313$	28.6198	1.61769	0.808844	−	0.588023i	$-0.200094\pi$
$313$	28.6198	1.61769	0.808844	+	0.588023i	$0.200094\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.65486	−0.0929462	−0.0464731	−	0.998920i	$-0.514798\pi$
$317$	−1.65486	−0.0929462	−0.0464731	+	0.998920i	$0.514798\pi$
$318$	0	0
$319$	1.38151	0.0773500
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.92101	0.329454
$324$	0	0
$325$	2.84787	0.157971
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.46050	0.245916
$330$	0	0
$331$	11.4284	0.628162	0.314081	−	0.949396i	$-0.398304\pi$
$331$	11.4284	0.628162	0.314081	+	0.949396i	$0.398304\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.6037	−0.743250
$336$	0	0
$337$	12.4179	0.676446	0.338223	−	0.941066i	$-0.390174\pi$
$337$	12.4179	0.676446	0.338223	+	0.941066i	$0.390174\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.5074	−0.569007
$342$	0	0
$343$	−19.5510	−1.05565
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.1331	−0.919752	−0.459876	−	0.887983i	$-0.652106\pi$
$347$	−17.1331	−0.919752	−0.459876	+	0.887983i	$0.652106\pi$
$348$	0	0
$349$	32.6811	1.74938	0.874690	−	0.484683i	$-0.161065\pi$
$349$	32.6811	1.74938	0.874690	+	0.484683i	$0.161065\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.9253	−1.11374	−0.556870	−	0.830600i	$-0.687998\pi$
$353$	−20.9253	−1.11374	−0.556870	+	0.830600i	$0.687998\pi$
$354$	0	0
$355$	−5.43852	−0.288647
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−29.0541	−1.53342	−0.766708	−	0.641996i	$-0.778107\pi$
$359$	−29.0541	−1.53342	−0.766708	+	0.641996i	$0.778107\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.83190	0.409940
$366$	0	0
$367$	0.165178	0.00862221	0.00431110	−	0.999991i	$-0.498628\pi$
$367$	0.165178	0.00862221	0.00431110	+	0.999991i	$0.498628\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.6840	−0.762357
$372$	0	0
$373$	−14.7807	−0.765318	−0.382659	−	0.923890i	$-0.624992\pi$
$373$	−14.7807	−0.765318	−0.382659	+	0.923890i	$0.624992\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.45038	0.177704
$378$	0	0
$379$	−25.3786	−1.30361	−0.651805	−	0.758386i	$-0.725988\pi$
$379$	−25.3786	−1.30361	−0.651805	+	0.758386i	$0.725988\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.29533	0.270579	0.135289	−	0.990806i	$-0.456804\pi$
$383$	5.29533	0.270579	0.135289	+	0.990806i	$0.456804\pi$
$384$	0	0
$385$	−7.38151	−0.376197
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.14027	0.362026	0.181013	−	0.983481i	$-0.442062\pi$
$389$	7.14027	0.362026	0.181013	+	0.983481i	$0.442062\pi$
$390$	0	0
$391$	−1.89610	−0.0958901
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.8889	−0.749142
$396$	0	0
$397$	6.74863	0.338704	0.169352	−	0.985556i	$-0.445832\pi$
$397$	6.74863	0.338704	0.169352	+	0.985556i	$0.445832\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.3609	−1.36634	−0.683169	−	0.730261i	$-0.739399\pi$
$401$	−27.3609	−1.36634	−0.683169	+	0.730261i	$0.739399\pi$
$402$	0	0
$403$	−26.2426	−1.30724
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.78794	−0.187761
$408$	0	0
$409$	4.45758	0.220413	0.110207	−	0.993909i	$-0.464849\pi$
$409$	4.45758	0.220413	0.110207	+	0.993909i	$0.464849\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	27.6768	1.36189
$414$	0	0
$415$	0.162253	0.00796467
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.4356	−1.78000	−0.889998	−	0.455965i	$-0.849294\pi$
$419$	−36.4356	−1.78000	−0.889998	+	0.455965i	$0.849294\pi$
$420$	0	0
$421$	11.2091	0.546300	0.273150	−	0.961971i	$-0.411934\pi$
$421$	11.2091	0.546300	0.273150	+	0.961971i	$0.411934\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.62276	0.224237
$426$	0	0
$427$	6.97509	0.337548
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.77023	0.229774	0.114887	−	0.993379i	$-0.463349\pi$
$431$	4.77023	0.229774	0.114887	+	0.993379i	$0.463349\pi$
$432$	0	0
$433$	−20.7381	−0.996611	−0.498305	−	0.867002i	$-0.666044\pi$
$433$	−20.7381	−0.996611	−0.498305	+	0.867002i	$0.666044\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.320233	−0.0153188
$438$	0	0
$439$	−2.46770	−0.117777	−0.0588885	−	0.998265i	$-0.518756\pi$
$439$	−2.46770	−0.117777	−0.0588885	+	0.998265i	$0.518756\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.7601	−1.31892	−0.659461	−	0.751739i	$-0.729216\pi$
$443$	−27.7601	−1.31892	−0.659461	+	0.751739i	$0.729216\pi$
$444$	0	0
$445$	−26.2016	−1.24207
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.0512	1.41820	0.709101	−	0.705107i	$-0.249101\pi$
$449$	30.0512	1.41820	0.709101	+	0.705107i	$0.249101\pi$
$450$	0	0
$451$	10.0292	0.472256
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.4356	−0.864274
$456$	0	0
$457$	37.4150	1.75020	0.875099	−	0.483944i	$-0.160796\pi$
$457$	37.4150	1.75020	0.875099	+	0.483944i	$0.160796\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.3815	1.04241	0.521206	−	0.853431i	$-0.325482\pi$
$461$	22.3815	1.04241	0.521206	+	0.853431i	$0.325482\pi$
$462$	0	0
$463$	15.5729	0.723736	0.361868	−	0.932229i	$-0.382139\pi$
$463$	15.5729	0.723736	0.361868	+	0.932229i	$0.382139\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.6870	−0.633357	−0.316679	−	0.948533i	$-0.602568\pi$
$467$	−13.6870	−0.633357	−0.316679	+	0.948533i	$0.602568\pi$
$468$	0	0
$469$	−16.2953	−0.752448
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.45331	0.388683
$474$	0	0
$475$	0.780738	0.0358227
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.7630	−1.22283	−0.611417	−	0.791309i	$-0.709400\pi$
$479$	−26.7630	−1.22283	−0.611417	+	0.791309i	$0.709400\pi$
$480$	0	0
$481$	−9.46050	−0.431362
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22.5586	−1.02433
$486$	0	0
$487$	−17.8200	−0.807503	−0.403752	−	0.914869i	$-0.632294\pi$
$487$	−17.8200	−0.807503	−0.403752	+	0.914869i	$0.632294\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.90662	0.221432	0.110716	−	0.993852i	$-0.464686\pi$
$491$	4.90662	0.221432	0.110716	+	0.993852i	$0.464686\pi$
$492$	0	0
$493$	5.60078	0.252246
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.51459	−0.292219
$498$	0	0
$499$	29.8023	1.33414	0.667068	−	0.744997i	$-0.267549\pi$
$499$	29.8023	1.33414	0.667068	+	0.744997i	$0.267549\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.67977	−0.119485	−0.0597424	−	0.998214i	$-0.519028\pi$
$503$	−2.67977	−0.119485	−0.0597424	+	0.998214i	$0.519028\pi$
$504$	0	0
$505$	0.399223	0.0177652
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.99280	0.442923	0.221462	−	0.975169i	$-0.428917\pi$
$509$	9.99280	0.442923	0.221462	+	0.975169i	$0.428917\pi$
$510$	0	0
$511$	9.38151	0.415014
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.71187	0.119499
$516$	0	0
$517$	−2.64766	−0.116444
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.6840	−1.56335	−0.781673	−	0.623688i	$-0.785633\pi$
$521$	−35.6840	−1.56335	−0.781673	+	0.623688i	$0.785633\pi$
$522$	0	0
$523$	−1.10097	−0.0481421	−0.0240711	−	0.999710i	$-0.507663\pi$
$523$	−1.10097	−0.0481421	−0.0240711	+	0.999710i	$0.507663\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.5979	−1.85559
$528$	0	0
$529$	−22.8975	−0.995541
$530$	0	0
$531$	0	0
$532$	0	0
$533$	25.0482	1.08496
$534$	0	0
$535$	−24.8200	−1.07306
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.38151	0.0595060
$540$	0	0
$541$	37.9105	1.62990	0.814950	−	0.579532i	$-0.196764\pi$
$541$	37.9105	1.62990	0.814950	+	0.579532i	$0.196764\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	29.0905	1.24610
$546$	0	0
$547$	22.4504	0.959909	0.479955	−	0.877293i	$-0.340653\pi$
$547$	22.4504	0.959909	0.479955	+	0.877293i	$0.340653\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.945916	0.0402974
$552$	0	0
$553$	−17.8348	−0.758414
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−41.6241	−1.76367	−0.881835	−	0.471558i	$-0.843692\pi$
$557$	−41.6241	−1.76367	−0.881835	+	0.471558i	$0.843692\pi$
$558$	0	0
$559$	21.1124	0.892961
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.0335	0.928599	0.464300	−	0.885678i	$-0.346306\pi$
$563$	22.0335	0.928599	0.464300	+	0.885678i	$0.346306\pi$
$564$	0	0
$565$	−19.7060	−0.829038
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.72665	−0.114307	−0.0571536	−	0.998365i	$-0.518202\pi$
$569$	−2.72665	−0.114307	−0.0571536	+	0.998365i	$0.518202\pi$
$570$	0	0
$571$	−8.28813	−0.346847	−0.173424	−	0.984847i	$-0.555483\pi$
$571$	−8.28813	−0.346847	−0.173424	+	0.984847i	$0.555483\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.250018	−0.0104265
$576$	0	0
$577$	−9.69455	−0.403589	−0.201795	−	0.979428i	$-0.564677\pi$
$577$	−9.69455	−0.403589	−0.201795	+	0.979428i	$0.564677\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.194356	0.00806324
$582$	0	0
$583$	8.71614	0.360986
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.9827	−0.494578	−0.247289	−	0.968942i	$-0.579540\pi$
$587$	−11.9827	−0.494578	−0.247289	+	0.968942i	$0.579540\pi$
$588$	0	0
$589$	−7.19436	−0.296438
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.6113	−0.600014	−0.300007	−	0.953937i	$-0.596989\pi$
$593$	−14.6113	−0.600014	−0.300007	+	0.953937i	$0.596989\pi$
$594$	0	0
$595$	−29.9253	−1.22682
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22.2599	0.909515	0.454758	−	0.890615i	$-0.349726\pi$
$599$	22.2599	0.909515	0.454758	+	0.890615i	$0.349726\pi$
$600$	0	0
$601$	40.0947	1.63550	0.817749	−	0.575575i	$-0.195222\pi$
$601$	40.0947	1.63550	0.817749	+	0.575575i	$0.195222\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−18.2134	−0.740480
$606$	0	0
$607$	−9.25564	−0.375675	−0.187837	−	0.982200i	$-0.560148\pi$
$607$	−9.25564	−0.375675	−0.187837	+	0.982200i	$0.560148\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.61264	−0.267519
$612$	0	0
$613$	16.4385	0.663946	0.331973	−	0.943289i	$-0.392286\pi$
$613$	16.4385	0.663946	0.331973	+	0.943289i	$0.392286\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−39.4868	−1.58968	−0.794839	−	0.606821i	$-0.792444\pi$
$617$	−39.4868	−1.58968	−0.794839	+	0.606821i	$0.792444\pi$
$618$	0	0
$619$	20.6228	0.828899	0.414449	−	0.910072i	$-0.363974\pi$
$619$	20.6228	0.828899	0.414449	+	0.910072i	$0.363974\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−31.3858	−1.25744
$624$	0	0
$625$	−20.4868	−0.819470
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−15.3566	−0.612308
$630$	0	0
$631$	26.6549	1.06111	0.530557	−	0.847649i	$-0.321983\pi$
$631$	26.6549	1.06111	0.530557	+	0.847649i	$0.321983\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	30.9311	1.22746
$636$	0	0
$637$	3.45038	0.136709
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.31304	0.288848	0.144424	−	0.989516i	$-0.453867\pi$
$641$	7.31304	0.288848	0.144424	+	0.989516i	$0.453867\pi$
$642$	0	0
$643$	−28.0115	−1.10466	−0.552332	−	0.833624i	$-0.686262\pi$
$643$	−28.0115	−1.10466	−0.552332	+	0.833624i	$0.686262\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.1288	−0.673403	−0.336701	−	0.941611i	$-0.609311\pi$
$647$	−17.1288	−0.673403	−0.336701	+	0.941611i	$0.609311\pi$
$648$	0	0
$649$	−16.4284	−0.644871
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.43560	0.134445	0.0672227	−	0.997738i	$-0.478586\pi$
$653$	3.43560	0.134445	0.0672227	+	0.997738i	$0.478586\pi$
$654$	0	0
$655$	19.3815	0.757298
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−29.6562	−1.15524	−0.577621	−	0.816305i	$-0.696019\pi$
$659$	−29.6562	−1.15524	−0.577621	+	0.816305i	$0.696019\pi$
$660$	0	0
$661$	−26.8214	−1.04323	−0.521615	−	0.853181i	$-0.674670\pi$
$661$	−26.8214	−1.04323	−0.521615	+	0.853181i	$0.674670\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.05408	−0.195989
$666$	0	0
$667$	−0.302914	−0.0117289
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.14027	−0.159833
$672$	0	0
$673$	−2.54377	−0.0980550	−0.0490275	−	0.998797i	$-0.515612\pi$
$673$	−2.54377	−0.0980550	−0.0490275	+	0.998797i	$0.515612\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.9224	−1.18844	−0.594221	−	0.804302i	$-0.702540\pi$
$677$	−30.9224	−1.18844	−0.594221	+	0.804302i	$0.702540\pi$
$678$	0	0
$679$	−27.0220	−1.03701
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.6549	1.09645	0.548224	−	0.836332i	$-0.315304\pi$
$683$	28.6549	1.09645	0.548224	+	0.836332i	$0.315304\pi$
$684$	0	0
$685$	−26.3638	−1.00731
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21.7689	0.829328
$690$	0	0
$691$	−18.0292	−0.685862	−0.342931	−	0.939361i	$-0.611420\pi$
$691$	−18.0292	−0.685862	−0.342931	+	0.939361i	$0.611420\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.8230	−0.562267
$696$	0	0
$697$	40.6591	1.54007
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14.5175	0.548319	0.274159	−	0.961684i	$-0.411600\pi$
$701$	14.5175	0.548319	0.274159	+	0.961684i	$0.411600\pi$
$702$	0	0
$703$	−2.59358	−0.0978187
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.478214	0.0179851
$708$	0	0
$709$	−39.6883	−1.49053	−0.745263	−	0.666771i	$-0.767676\pi$
$709$	−39.6883	−1.49053	−0.745263	+	0.666771i	$0.767676\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.30387	0.0862807
$714$	0	0
$715$	10.9430	0.409245
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.33463	−0.0497732	−0.0248866	−	0.999690i	$-0.507922\pi$
$719$	−1.33463	−0.0497732	−0.0248866	+	0.999690i	$0.507922\pi$
$720$	0	0
$721$	3.24844	0.120978
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.738512	0.0274277
$726$	0	0
$727$	9.17996	0.340466	0.170233	−	0.985404i	$-0.445548\pi$
$727$	9.17996	0.340466	0.170233	+	0.985404i	$0.445548\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	34.2704	1.26754
$732$	0	0
$733$	1.97217	0.0728437	0.0364219	−	0.999337i	$-0.488404\pi$
$733$	1.97217	0.0728437	0.0364219	+	0.999337i	$0.488404\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.67257	0.356294
$738$	0	0
$739$	12.5471	0.461552	0.230776	−	0.973007i	$-0.425874\pi$
$739$	12.5471	0.461552	0.230776	+	0.973007i	$0.425874\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.61849	0.169436	0.0847179	−	0.996405i	$-0.473001\pi$
$743$	4.61849	0.169436	0.0847179	+	0.996405i	$0.473001\pi$
$744$	0	0
$745$	7.71480	0.282648
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−29.7309	−1.08634
$750$	0	0
$751$	−19.7951	−0.722335	−0.361167	−	0.932501i	$-0.617622\pi$
$751$	−19.7951	−0.722335	−0.361167	+	0.932501i	$0.617622\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−14.8889	−0.541863
$756$	0	0
$757$	2.08950	0.0759442	0.0379721	−	0.999279i	$-0.487910\pi$
$757$	2.08950	0.0759442	0.0379721	+	0.999279i	$0.487910\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.6021	−1.10933	−0.554663	−	0.832075i	$-0.687153\pi$
$761$	−30.6021	−1.10933	−0.554663	+	0.832075i	$0.687153\pi$
$762$	0	0
$763$	34.8463	1.26152
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−41.0305	−1.48153
$768$	0	0
$769$	3.65913	0.131952	0.0659759	−	0.997821i	$-0.478984\pi$
$769$	3.65913	0.131952	0.0659759	+	0.997821i	$0.478984\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.5801	1.13586	0.567929	−	0.823077i	$-0.307745\pi$
$773$	31.5801	1.13586	0.567929	+	0.823077i	$0.307745\pi$
$774$	0	0
$775$	−5.61691	−0.201765
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.86693	0.246033
$780$	0	0
$781$	3.86693	0.138369
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.2556	−0.687263
$786$	0	0
$787$	−14.9780	−0.533909	−0.266954	−	0.963709i	$-0.586017\pi$
$787$	−14.9780	−0.533909	−0.266954	+	0.963709i	$0.586017\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−23.6050	−0.839299
$792$	0	0
$793$	−10.3405	−0.367201
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.5175	−0.833033	−0.416517	−	0.909128i	$-0.636749\pi$
$797$	−23.5175	−0.833033	−0.416517	+	0.909128i	$0.636749\pi$
$798$	0	0
$799$	−10.7339	−0.379736
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.56867	−0.196514
$804$	0	0
$805$	1.61849	0.0570441
$806$	0	0
$807$	0	0
$808$	0	0
$809$	25.1593	0.884555	0.442277	−	0.896878i	$-0.354171\pi$
$809$	25.1593	0.884555	0.442277	+	0.896878i	$0.354171\pi$
$810$	0	0
$811$	4.49300	0.157770	0.0788852	−	0.996884i	$-0.474864\pi$
$811$	4.49300	0.157770	0.0788852	+	0.996884i	$0.474864\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−43.6912	−1.53044
$816$	0	0
$817$	5.78794	0.202494
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−45.5831	−1.59086	−0.795430	−	0.606045i	$-0.792755\pi$
$821$	−45.5831	−1.59086	−0.795430	+	0.606045i	$0.792755\pi$
$822$	0	0
$823$	−33.2130	−1.15773	−0.578867	−	0.815422i	$-0.696505\pi$
$823$	−33.2130	−1.15773	−0.578867	+	0.815422i	$0.696505\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.0364	1.46175	0.730874	−	0.682512i	$-0.239113\pi$
$827$	42.0364	1.46175	0.730874	+	0.682512i	$0.239113\pi$
$828$	0	0
$829$	−19.1838	−0.666282	−0.333141	−	0.942877i	$-0.608109\pi$
$829$	−19.1838	−0.666282	−0.333141	+	0.942877i	$0.608109\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.60078	0.194055
$834$	0	0
$835$	−8.66672	−0.299924
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.3432	1.39280	0.696400	−	0.717653i	$-0.254784\pi$
$839$	40.3432	1.39280	0.696400	+	0.717653i	$0.254784\pi$
$840$	0	0
$841$	−28.1052	−0.969146
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.627419	0.0215839
$846$	0	0
$847$	−21.8171	−0.749645
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.830550	0.0284709
$852$	0	0
$853$	41.4251	1.41837	0.709184	−	0.705023i	$-0.249064\pi$
$853$	41.4251	1.41837	0.709184	+	0.705023i	$0.249064\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.4543	0.869501	0.434751	−	0.900551i	$-0.356837\pi$
$857$	25.4543	0.869501	0.434751	+	0.900551i	$0.356837\pi$
$858$	0	0
$859$	−43.0410	−1.46854	−0.734271	−	0.678857i	$-0.762476\pi$
$859$	−43.0410	−1.46854	−0.734271	+	0.678857i	$0.762476\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	53.6050	1.82474	0.912369	−	0.409370i	$-0.134251\pi$
$863$	53.6050	1.82474	0.912369	+	0.409370i	$0.134251\pi$
$864$	0	0
$865$	−7.54377	−0.256496
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10.5864	0.359118
$870$	0	0
$871$	24.1576	0.818549
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−29.2163	−0.987692
$876$	0	0
$877$	10.2311	0.345480	0.172740	−	0.984967i	$-0.444738\pi$
$877$	10.2311	0.345480	0.172740	+	0.984967i	$0.444738\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−27.3393	−0.921084	−0.460542	−	0.887638i	$-0.652345\pi$
$881$	−27.3393	−0.921084	−0.460542	+	0.887638i	$0.652345\pi$
$882$	0	0
$883$	17.9899	0.605408	0.302704	−	0.953085i	$-0.402111\pi$
$883$	17.9899	0.605408	0.302704	+	0.953085i	$0.402111\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.0187	1.34370	0.671848	−	0.740689i	$-0.265501\pi$
$887$	40.0187	1.34370	0.671848	+	0.740689i	$0.265501\pi$
$888$	0	0
$889$	37.0512	1.24266
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.81284	−0.0606644
$894$	0	0
$895$	−34.4475	−1.15145
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.80525	−0.226968
$900$	0	0
$901$	35.3360	1.17721
$902$	0	0
$903$	0	0
$904$	0	0
$905$	42.4720	1.41182
$906$	0	0
$907$	3.69455	0.122676	0.0613378	−	0.998117i	$-0.480463\pi$
$907$	3.69455	0.122676	0.0613378	+	0.998117i	$0.480463\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.7021	0.420841	0.210420	−	0.977611i	$-0.432517\pi$
$911$	12.7021	0.420841	0.210420	+	0.977611i	$0.432517\pi$
$912$	0	0
$913$	−0.115366	−0.00381805
$914$	0	0
$915$	0	0
$916$	0	0
$917$	23.2163	0.766671
$918$	0	0
$919$	41.9177	1.38274	0.691369	−	0.722502i	$-0.257008\pi$
$919$	41.9177	1.38274	0.691369	+	0.722502i	$0.257008\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.65779	0.317890
$924$	0	0
$925$	−2.02491	−0.0665785
$926$	0	0
$927$	0	0
$928$	0	0
$929$	59.8653	1.96412	0.982059	−	0.188573i	$-0.0603862\pi$
$929$	59.8653	1.96412	0.982059	+	0.188573i	$0.0603862\pi$
$930$	0	0
$931$	0.945916	0.0310011
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17.7630	0.580913
$936$	0	0
$937$	−0.194745	−0.00636205	−0.00318102	−	0.999995i	$-0.501013\pi$
$937$	−0.194745	−0.00636205	−0.00318102	+	0.999995i	$0.501013\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−9.73424	−0.317327	−0.158664	−	0.987333i	$-0.550719\pi$
$941$	−9.73424	−0.317327	−0.158664	+	0.987333i	$0.550719\pi$
$942$	0	0
$943$	−2.19902	−0.0716099
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35.2780	1.14638	0.573191	−	0.819422i	$-0.305705\pi$
$947$	35.2780	1.14638	0.573191	+	0.819422i	$0.305705\pi$
$948$	0	0
$949$	−13.9080	−0.451472
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.311689	−0.0100966	−0.00504830	−	0.999987i	$-0.501607\pi$
$953$	−0.311689	−0.0100966	−0.00504830	+	0.999987i	$0.501607\pi$
$954$	0	0
$955$	−34.9646	−1.13143
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−31.5801	−1.01978
$960$	0	0
$961$	20.7588	0.669637
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14.9401	−0.480938
$966$	0	0
$967$	−22.8204	−0.733855	−0.366928	−	0.930249i	$-0.619590\pi$
$967$	−22.8204	−0.733855	−0.366928	+	0.930249i	$0.619590\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53.1200	1.70470	0.852351	−	0.522969i	$-0.175176\pi$
$971$	53.1200	1.70470	0.852351	+	0.522969i	$0.175176\pi$
$972$	0	0
$973$	−17.7558	−0.569226
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.7444	−0.439721	−0.219861	−	0.975531i	$-0.570560\pi$
$977$	−13.7444	−0.439721	−0.219861	+	0.975531i	$0.570560\pi$
$978$	0	0
$979$	18.6300	0.595416
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35.7395	1.13991	0.569956	−	0.821675i	$-0.306960\pi$
$983$	35.7395	1.13991	0.569956	+	0.821675i	$0.306960\pi$
$984$	0	0
$985$	−52.8683	−1.68452
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.85349	−0.0589375
$990$	0	0
$991$	39.0220	1.23957	0.619787	−	0.784770i	$-0.287219\pi$
$991$	39.0220	1.23957	0.619787	+	0.784770i	$0.287219\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	32.7119	1.03704
$996$	0	0
$997$	−10.1623	−0.321842	−0.160921	−	0.986967i	$-0.551446\pi$
$997$	−10.1623	−0.321842	−0.160921	+	0.986967i	$0.551446\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.bg.1.3		3
3.2	odd	2		8208.2.a.bm.1.1		3
4.3	odd	2		2052.2.a.e.1.3	✓	3
12.11	even	2		2052.2.a.f.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2052.2.a.e.1.3	✓	3	4.3	odd	2
2052.2.a.f.1.1	yes	3	12.11	even	2
8208.2.a.bg.1.3		3	1.1	even	1	trivial
8208.2.a.bm.1.1		3	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05408 q^{5} +2.46050 q^{7} -1.46050 q^{11} -3.64766 q^{13} -5.92101 q^{17} -1.00000 q^{19} +0.320233 q^{23} -0.780738 q^{25} -0.945916 q^{29} +7.19436 q^{31} +5.05408 q^{35} +2.59358 q^{37} -6.86693 q^{41} -5.78794 q^{43} +1.81284 q^{47} -0.945916 q^{49} -5.96790 q^{53} -3.00000 q^{55} +11.2484 q^{59} +2.83482 q^{61} -7.49261 q^{65} -6.62276 q^{67} -2.64766 q^{71} +3.81284 q^{73} -3.59358 q^{77} -7.24844 q^{79} +0.0789903 q^{83} -12.1623 q^{85} -12.7558 q^{89} -8.97509 q^{91} -2.05408 q^{95} -10.9823 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + q^{7} + 2 q^{11} + q^{13} - 5 q^{17} - 3 q^{19} - q^{23} + 6 q^{25} - 12 q^{29} + 8 q^{31} + 6 q^{35} + 5 q^{37} - 17 q^{41} - q^{43} + 11 q^{47} - 12 q^{49} - 4 q^{53} - 9 q^{55} + 11 q^{59}+ \cdots - 3 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + q^7 + 2 * q^11 + q^13 - 5 * q^17 - 3 * q^19 - q^23 + 6 * q^25 - 12 * q^29 + 8 * q^31 + 6 * q^35 + 5 * q^37 - 17 * q^41 - q^43 + 11 * q^47 - 12 * q^49 - 4 * q^53 - 9 * q^55 + 11 * q^59 - 9 * q^61 - 30 * q^65 + 14 * q^67 + 4 * q^71 + 17 * q^73 - 8 * q^77 + q^79 + 13 * q^83 - 9 * q^85 - 8 * q^89 - 5 * q^91 + 3 * q^95 - 3 * q^97

Introduction

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