LMFDB - Embedded newform 8208.2.a.bj.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,-1,0,3,0,0,0,0] 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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.69963$ 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)

$f(q)$	$=$	$q+1.58836 q^{5} +1.11126 q^{7} -0.111264 q^{11} -0.888736 q^{13} +1.00000 q^{17} -1.00000 q^{19} +4.98762 q^{23} -2.47710 q^{25} -9.98762 q^{29} -1.51052 q^{31} +1.76509 q^{35} -3.69963 q^{37} -8.38688 q^{41} -4.98762 q^{43} -5.79851 q^{47} -5.76509 q^{49} +2.69963 q^{53} -0.176728 q^{55} -12.2756 q^{59} -8.28799 q^{61} -1.41164 q^{65} +12.6749 q^{67} -4.33379 q^{71} -3.60074 q^{73} -0.123644 q^{77} +2.74543 q^{79} +6.97524 q^{83} +1.58836 q^{85} +14.3090 q^{89} -0.987620 q^{91} -1.58836 q^{95} +15.0741 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{5} + 3 q^{7} - 3 q^{13} + 3 q^{17} - 3 q^{19} - 3 q^{23} - 2 q^{25} - 12 q^{29} + 8 q^{31} - 12 q^{35} - 5 q^{37} + 5 q^{41} + 3 q^{43} + 7 q^{47} + 2 q^{53} + 11 q^{55} - 7 q^{59} - 13 q^{61}+ \cdots - 9 q^{97}+O(q^{100}) $ 3 * q - q^5 + 3 * q^7 - 3 * q^13 + 3 * q^17 - 3 * q^19 - 3 * q^23 - 2 * q^25 - 12 * q^29 + 8 * q^31 - 12 * q^35 - 5 * q^37 + 5 * q^41 + 3 * q^43 + 7 * q^47 + 2 * q^53 + 11 * q^55 - 7 * q^59 - 13 * q^61 - 10 * q^65 - 4 * q^67 - 12 * q^71 - 23 * q^73 - 18 * q^77 + 13 * q^79 - 15 * q^83 - q^85 + 6 * q^89 + 15 * q^91 + q^95 - 9 * 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
$5$	1.58836	0.710338	0.355169	−	0.934802i	$-0.384423\pi$
$5$	1.58836	0.710338	0.355169	+	0.934802i	$0.384423\pi$
$6$	0	0
$7$	1.11126	0.420018	0.210009	−	0.977699i	$-0.432651\pi$
$7$	1.11126	0.420018	0.210009	+	0.977699i	$0.432651\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.111264	−0.0335474	−0.0167737	−	0.999859i	$-0.505339\pi$
$11$	−0.111264	−0.0335474	−0.0167737	+	0.999859i	$0.505339\pi$
$12$	0	0
$13$	−0.888736	−0.246491	−0.123245	−	0.992376i	$-0.539330\pi$
$13$	−0.888736	−0.246491	−0.123245	+	0.992376i	$0.539330\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.98762	1.03999	0.519995	−	0.854169i	$-0.325934\pi$
$23$	4.98762	1.03999	0.519995	+	0.854169i	$0.325934\pi$
$24$	0	0
$25$	−2.47710	−0.495420
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.98762	−1.85465	−0.927327	−	0.374251i	$-0.877900\pi$
$29$	−9.98762	−1.85465	−0.927327	+	0.374251i	$0.877900\pi$
$30$	0	0
$31$	−1.51052	−0.271297	−0.135649	−	0.990757i	$-0.543312\pi$
$31$	−1.51052	−0.271297	−0.135649	+	0.990757i	$0.543312\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.76509	0.298355
$36$	0	0
$37$	−3.69963	−0.608215	−0.304108	−	0.952638i	$-0.598358\pi$
$37$	−3.69963	−0.608215	−0.304108	+	0.952638i	$0.598358\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.38688	−1.30981	−0.654905	−	0.755711i	$-0.727291\pi$
$41$	−8.38688	−1.30981	−0.654905	+	0.755711i	$0.727291\pi$
$42$	0	0
$43$	−4.98762	−0.760605	−0.380302	−	0.924862i	$-0.624180\pi$
$43$	−4.98762	−0.760605	−0.380302	+	0.924862i	$0.624180\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.79851	−0.845800	−0.422900	−	0.906176i	$-0.638988\pi$
$47$	−5.79851	−0.845800	−0.422900	+	0.906176i	$0.638988\pi$
$48$	0	0
$49$	−5.76509	−0.823585
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.69963	0.370822	0.185411	−	0.982661i	$-0.440638\pi$
$53$	2.69963	0.370822	0.185411	+	0.982661i	$0.440638\pi$
$54$	0	0
$55$	−0.176728	−0.0238300
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.2756	−1.59815	−0.799074	−	0.601232i	$-0.794677\pi$
$59$	−12.2756	−1.59815	−0.799074	+	0.601232i	$0.794677\pi$
$60$	0	0
$61$	−8.28799	−1.06117	−0.530584	−	0.847632i	$-0.678027\pi$
$61$	−8.28799	−1.06117	−0.530584	+	0.847632i	$0.678027\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.41164	−0.175092
$66$	0	0
$67$	12.6749	1.54848	0.774241	−	0.632891i	$-0.218132\pi$
$67$	12.6749	1.54848	0.774241	+	0.632891i	$0.218132\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.33379	−0.514327	−0.257163	−	0.966368i	$-0.582788\pi$
$71$	−4.33379	−0.514327	−0.257163	+	0.966368i	$0.582788\pi$
$72$	0	0
$73$	−3.60074	−0.421435	−0.210718	−	0.977547i	$-0.567580\pi$
$73$	−3.60074	−0.421435	−0.210718	+	0.977547i	$0.567580\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.123644	−0.0140905
$78$	0	0
$79$	2.74543	0.308885	0.154442	−	0.988002i	$-0.450642\pi$
$79$	2.74543	0.308885	0.154442	+	0.988002i	$0.450642\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.97524	0.765632	0.382816	−	0.923825i	$-0.374954\pi$
$83$	6.97524	0.765632	0.382816	+	0.923825i	$0.374954\pi$
$84$	0	0
$85$	1.58836	0.172282
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.3090	1.51675	0.758377	−	0.651816i	$-0.225993\pi$
$89$	14.3090	1.51675	0.758377	+	0.651816i	$0.225993\pi$
$90$	0	0
$91$	−0.987620	−0.103531
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.58836	−0.162963
$96$	0	0
$97$	15.0741	1.53055	0.765273	−	0.643706i	$-0.222604\pi$
$97$	15.0741	1.53055	0.765273	+	0.643706i	$0.222604\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.51052	−0.846828	−0.423414	−	0.905936i	$-0.639168\pi$
$101$	−8.51052	−0.846828	−0.423414	+	0.905936i	$0.639168\pi$
$102$	0	0
$103$	8.36584	0.824310	0.412155	−	0.911114i	$-0.364776\pi$
$103$	8.36584	0.824310	0.412155	+	0.911114i	$0.364776\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.01238	−0.194544	−0.0972720	−	0.995258i	$-0.531012\pi$
$107$	−2.01238	−0.194544	−0.0972720	+	0.995258i	$0.531012\pi$
$108$	0	0
$109$	−13.4327	−1.28662	−0.643309	−	0.765607i	$-0.722439\pi$
$109$	−13.4327	−1.28662	−0.643309	+	0.765607i	$0.722439\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.65383	−0.908156	−0.454078	−	0.890962i	$-0.650031\pi$
$113$	−9.65383	−0.908156	−0.454078	+	0.890962i	$0.650031\pi$
$114$	0	0
$115$	7.92216	0.738745
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.11126	0.101869
$120$	0	0
$121$	−10.9876	−0.998875
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.8764	−1.06225
$126$	0	0
$127$	10.0210	0.889224	0.444612	−	0.895723i	$-0.353342\pi$
$127$	10.0210	0.889224	0.444612	+	0.895723i	$0.353342\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.6080	−0.926828	−0.463414	−	0.886142i	$-0.653376\pi$
$131$	−10.6080	−0.926828	−0.463414	+	0.886142i	$0.653376\pi$
$132$	0	0
$133$	−1.11126	−0.0963588
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.31275	−0.368463	−0.184232	−	0.982883i	$-0.558980\pi$
$137$	−4.31275	−0.368463	−0.184232	+	0.982883i	$0.558980\pi$
$138$	0	0
$139$	1.02476	0.0869190	0.0434595	−	0.999055i	$-0.486162\pi$
$139$	1.02476	0.0869190	0.0434595	+	0.999055i	$0.486162\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.0988844	0.00826913
$144$	0	0
$145$	−15.8640	−1.31743
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.88874	0.236654	0.118327	−	0.992975i	$-0.462247\pi$
$149$	2.88874	0.236654	0.118327	+	0.992975i	$0.462247\pi$
$150$	0	0
$151$	−4.05308	−0.329835	−0.164918	−	0.986307i	$-0.552736\pi$
$151$	−4.05308	−0.329835	−0.164918	+	0.986307i	$0.552736\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.39926	−0.192713
$156$	0	0
$157$	−12.1520	−0.969833	−0.484916	−	0.874561i	$-0.661150\pi$
$157$	−12.1520	−0.969833	−0.484916	+	0.874561i	$0.661150\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.54256	0.436815
$162$	0	0
$163$	1.92078	0.150447	0.0752235	−	0.997167i	$-0.476033\pi$
$163$	1.92078	0.150447	0.0752235	+	0.997167i	$0.476033\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.47710	−0.114301	−0.0571507	−	0.998366i	$-0.518202\pi$
$167$	−1.47710	−0.114301	−0.0571507	+	0.998366i	$0.518202\pi$
$168$	0	0
$169$	−12.2101	−0.939242
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.45234	0.718648	0.359324	−	0.933213i	$-0.383007\pi$
$173$	9.45234	0.718648	0.359324	+	0.933213i	$0.383007\pi$
$174$	0	0
$175$	−2.75271	−0.208085
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.6414	−1.46807	−0.734035	−	0.679111i	$-0.762365\pi$
$179$	−19.6414	−1.46807	−0.734035	+	0.679111i	$0.762365\pi$
$180$	0	0
$181$	19.4400	1.44496	0.722480	−	0.691391i	$-0.243002\pi$
$181$	19.4400	1.44496	0.722480	+	0.691391i	$0.243002\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.87636	−0.432038
$186$	0	0
$187$	−0.111264	−0.00813644
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.93454	0.357051	0.178525	−	0.983935i	$-0.442867\pi$
$191$	4.93454	0.357051	0.178525	+	0.983935i	$0.442867\pi$
$192$	0	0
$193$	15.0197	1.08114	0.540570	−	0.841299i	$-0.318209\pi$
$193$	15.0197	1.08114	0.540570	+	0.841299i	$0.318209\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.8319	−0.914237	−0.457119	−	0.889406i	$-0.651119\pi$
$197$	−12.8319	−0.914237	−0.457119	+	0.889406i	$0.651119\pi$
$198$	0	0
$199$	3.03342	0.215033	0.107517	−	0.994203i	$-0.465710\pi$
$199$	3.03342	0.215033	0.107517	+	0.994203i	$0.465710\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−11.0989	−0.778989
$204$	0	0
$205$	−13.3214	−0.930408
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.111264	0.00769630
$210$	0	0
$211$	26.2051	1.80403	0.902015	−	0.431704i	$-0.142088\pi$
$211$	26.2051	1.80403	0.902015	+	0.431704i	$0.142088\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.92216	−0.540287
$216$	0	0
$217$	−1.67859	−0.113950
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.888736	−0.0597828
$222$	0	0
$223$	4.87636	0.326545	0.163272	−	0.986581i	$-0.447795\pi$
$223$	4.87636	0.326545	0.163272	+	0.986581i	$0.447795\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.76647	0.183617	0.0918085	−	0.995777i	$-0.470735\pi$
$227$	2.76647	0.183617	0.0918085	+	0.995777i	$0.470735\pi$
$228$	0	0
$229$	−8.41164	−0.555857	−0.277928	−	0.960602i	$-0.589648\pi$
$229$	−8.41164	−0.555857	−0.277928	+	0.960602i	$0.589648\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.33379	0.545965	0.272982	−	0.962019i	$-0.411990\pi$
$233$	8.33379	0.545965	0.272982	+	0.962019i	$0.411990\pi$
$234$	0	0
$235$	−9.21015	−0.600804
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.67487	0.496446	0.248223	−	0.968703i	$-0.420153\pi$
$239$	7.67487	0.496446	0.248223	+	0.968703i	$0.420153\pi$
$240$	0	0
$241$	−18.2101	−1.17302	−0.586509	−	0.809942i	$-0.699498\pi$
$241$	−18.2101	−1.17302	−0.586509	+	0.809942i	$0.699498\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.15706	−0.585023
$246$	0	0
$247$	0.888736	0.0565489
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−16.2880	−1.02809	−0.514044	−	0.857764i	$-0.671853\pi$
$251$	−16.2880	−1.02809	−0.514044	+	0.857764i	$0.671853\pi$
$252$	0	0
$253$	−0.554943	−0.0348890
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.41026	−0.0879695	−0.0439848	−	0.999032i	$-0.514005\pi$
$257$	−1.41026	−0.0879695	−0.0439848	+	0.999032i	$0.514005\pi$
$258$	0	0
$259$	−4.11126	−0.255462
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.56732	0.281633	0.140817	−	0.990036i	$-0.455027\pi$
$263$	4.56732	0.281633	0.140817	+	0.990036i	$0.455027\pi$
$264$	0	0
$265$	4.28799	0.263409
$266$	0	0
$267$	0	0
$268$	0	0
$269$	23.3942	1.42637	0.713184	−	0.700977i	$-0.247253\pi$
$269$	23.3942	1.42637	0.713184	+	0.700977i	$0.247253\pi$
$270$	0	0
$271$	−16.6428	−1.01098	−0.505490	−	0.862833i	$-0.668688\pi$
$271$	−16.6428	−1.01098	−0.505490	+	0.862833i	$0.668688\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.275612	0.0166201
$276$	0	0
$277$	12.4734	0.749453	0.374726	−	0.927135i	$-0.377737\pi$
$277$	12.4734	0.749453	0.374726	+	0.927135i	$0.377737\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.5760	−0.690565	−0.345283	−	0.938499i	$-0.612217\pi$
$281$	−11.5760	−0.690565	−0.345283	+	0.938499i	$0.612217\pi$
$282$	0	0
$283$	17.3993	1.03428	0.517139	−	0.855901i	$-0.326997\pi$
$283$	17.3993	1.03428	0.517139	+	0.855901i	$0.326997\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.32004	−0.550144
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.22981	0.422370	0.211185	−	0.977446i	$-0.432268\pi$
$293$	7.22981	0.422370	0.211185	+	0.977446i	$0.432268\pi$
$294$	0	0
$295$	−19.4981	−1.13523
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.43268	−0.256348
$300$	0	0
$301$	−5.54256	−0.319468
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−13.1643	−0.753788
$306$	0	0
$307$	30.9505	1.76644	0.883219	−	0.468961i	$-0.155372\pi$
$307$	30.9505	1.76644	0.883219	+	0.468961i	$0.155372\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.10026	−0.516029	−0.258014	−	0.966141i	$-0.583068\pi$
$311$	−9.10026	−0.516029	−0.258014	+	0.966141i	$0.583068\pi$
$312$	0	0
$313$	4.49086	0.253838	0.126919	−	0.991913i	$-0.459491\pi$
$313$	4.49086	0.253838	0.126919	+	0.991913i	$0.459491\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.8378	−1.78819	−0.894096	−	0.447876i	$-0.852181\pi$
$317$	−31.8378	−1.78819	−0.894096	+	0.447876i	$0.852181\pi$
$318$	0	0
$319$	1.11126	0.0622188
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.00000	−0.0556415
$324$	0	0
$325$	2.20149	0.122117
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.44368	−0.355252
$330$	0	0
$331$	33.0480	1.81648	0.908241	−	0.418448i	$-0.137426\pi$
$331$	33.0480	1.81648	0.908241	+	0.418448i	$0.137426\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	20.1323	1.09995
$336$	0	0
$337$	22.7280	1.23807	0.619035	−	0.785363i	$-0.287524\pi$
$337$	22.7280	1.23807	0.619035	+	0.785363i	$0.287524\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.168067	0.00910133
$342$	0	0
$343$	−14.1854	−0.765939
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.00866	0.322562	0.161281	−	0.986909i	$-0.448437\pi$
$347$	6.00866	0.322562	0.161281	+	0.986909i	$0.448437\pi$
$348$	0	0
$349$	−9.07784	−0.485926	−0.242963	−	0.970036i	$-0.578119\pi$
$349$	−9.07784	−0.485926	−0.242963	+	0.970036i	$0.578119\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.1891	−0.648761	−0.324380	−	0.945927i	$-0.605156\pi$
$353$	−12.1891	−0.648761	−0.324380	+	0.945927i	$0.605156\pi$
$354$	0	0
$355$	−6.88364	−0.365346
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.41164	−0.180059	−0.0900296	−	0.995939i	$-0.528696\pi$
$359$	−3.41164	−0.180059	−0.0900296	+	0.995939i	$0.528696\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.71929	−0.299361
$366$	0	0
$367$	11.8640	0.619295	0.309647	−	0.950851i	$-0.399789\pi$
$367$	11.8640	0.619295	0.309647	+	0.950851i	$0.399789\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.00000	0.155752
$372$	0	0
$373$	30.7417	1.59175	0.795873	−	0.605464i	$-0.207012\pi$
$373$	30.7417	1.59175	0.795873	+	0.605464i	$0.207012\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.87636	0.457156
$378$	0	0
$379$	16.6945	0.857541	0.428770	−	0.903414i	$-0.358947\pi$
$379$	16.6945	0.857541	0.428770	+	0.903414i	$0.358947\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.7280	−1.00805	−0.504026	−	0.863689i	$-0.668148\pi$
$383$	−19.7280	−1.00805	−0.504026	+	0.863689i	$0.668148\pi$
$384$	0	0
$385$	−0.196391	−0.0100090
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−23.2298	−1.17780	−0.588899	−	0.808207i	$-0.700438\pi$
$389$	−23.2298	−1.17780	−0.588899	+	0.808207i	$0.700438\pi$
$390$	0	0
$391$	4.98762	0.252235
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.36074	0.219413
$396$	0	0
$397$	−2.44506	−0.122714	−0.0613569	−	0.998116i	$-0.519543\pi$
$397$	−2.44506	−0.122714	−0.0613569	+	0.998116i	$0.519543\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.28799	0.164194	0.0820972	−	0.996624i	$-0.473838\pi$
$401$	3.28799	0.164194	0.0820972	+	0.996624i	$0.473838\pi$
$402$	0	0
$403$	1.34245	0.0668724
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.411636	0.0204040
$408$	0	0
$409$	−0.0544615	−0.00269295	−0.00134647	−	0.999999i	$-0.500429\pi$
$409$	−0.0544615	−0.00269295	−0.00134647	+	0.999999i	$0.500429\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.6414	−0.671252
$414$	0	0
$415$	11.0792	0.543858
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.9011	−1.21650	−0.608250	−	0.793746i	$-0.708128\pi$
$419$	−24.9011	−1.21650	−0.608250	+	0.793746i	$0.708128\pi$
$420$	0	0
$421$	−8.15706	−0.397551	−0.198775	−	0.980045i	$-0.563696\pi$
$421$	−8.15706	−0.397551	−0.198775	+	0.980045i	$0.563696\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.47710	−0.120157
$426$	0	0
$427$	−9.21015	−0.445710
$428$	0	0
$429$	0	0
$430$	0	0
$431$	38.8355	1.87064	0.935320	−	0.353803i	$-0.115112\pi$
$431$	38.8355	1.87064	0.935320	+	0.353803i	$0.115112\pi$
$432$	0	0
$433$	−4.27699	−0.205539	−0.102770	−	0.994705i	$-0.532770\pi$
$433$	−4.27699	−0.205539	−0.102770	+	0.994705i	$0.532770\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.98762	−0.238590
$438$	0	0
$439$	−21.4844	−1.02539	−0.512697	−	0.858570i	$-0.671354\pi$
$439$	−21.4844	−1.02539	−0.512697	+	0.858570i	$0.671354\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.7207	−1.31705	−0.658524	−	0.752559i	$-0.728819\pi$
$443$	−27.7207	−1.31705	−0.658524	+	0.752559i	$0.728819\pi$
$444$	0	0
$445$	22.7280	1.07741
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.37959	−0.0651070	−0.0325535	−	0.999470i	$-0.510364\pi$
$449$	−1.37959	−0.0651070	−0.0325535	+	0.999470i	$0.510364\pi$
$450$	0	0
$451$	0.933159	0.0439407
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.56870	−0.0735418
$456$	0	0
$457$	−6.14468	−0.287436	−0.143718	−	0.989619i	$-0.545906\pi$
$457$	−6.14468	−0.287436	−0.143718	+	0.989619i	$0.545906\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.8887	−1.11261	−0.556305	−	0.830978i	$-0.687781\pi$
$461$	−23.8887	−1.11261	−0.556305	+	0.830978i	$0.687781\pi$
$462$	0	0
$463$	13.4523	0.625183	0.312592	−	0.949888i	$-0.398803\pi$
$463$	13.4523	0.625183	0.312592	+	0.949888i	$0.398803\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.1868	−1.67452	−0.837262	−	0.546802i	$-0.815845\pi$
$467$	−36.1868	−1.67452	−0.837262	+	0.546802i	$0.815845\pi$
$468$	0	0
$469$	14.0851	0.650391
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.554943	0.0255163
$474$	0	0
$475$	2.47710	0.113657
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.57598	0.391847	0.195923	−	0.980619i	$-0.437230\pi$
$479$	8.57598	0.391847	0.195923	+	0.980619i	$0.437230\pi$
$480$	0	0
$481$	3.28799	0.149920
$482$	0	0
$483$	0	0
$484$	0	0
$485$	23.9432	1.08720
$486$	0	0
$487$	6.73305	0.305104	0.152552	−	0.988295i	$-0.451251\pi$
$487$	6.73305	0.305104	0.152552	+	0.988295i	$0.451251\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.55494	0.0701736	0.0350868	−	0.999384i	$-0.488829\pi$
$491$	1.55494	0.0701736	0.0350868	+	0.999384i	$0.488829\pi$
$492$	0	0
$493$	−9.98762	−0.449820
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.81599	−0.216027
$498$	0	0
$499$	−29.0938	−1.30242	−0.651208	−	0.758899i	$-0.725738\pi$
$499$	−29.0938	−1.30242	−0.651208	+	0.758899i	$0.725738\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.67996	0.119494	0.0597469	−	0.998214i	$-0.480971\pi$
$503$	2.67996	0.119494	0.0597469	+	0.998214i	$0.480971\pi$
$504$	0	0
$505$	−13.5178	−0.601534
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.6232	−0.470863	−0.235432	−	0.971891i	$-0.575650\pi$
$509$	−10.6232	−0.470863	−0.235432	+	0.971891i	$0.575650\pi$
$510$	0	0
$511$	−4.00138	−0.177011
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.2880	0.585539
$516$	0	0
$517$	0.645167	0.0283744
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.7069	0.556700	0.278350	−	0.960480i	$-0.410213\pi$
$521$	12.7069	0.556700	0.278350	+	0.960480i	$0.410213\pi$
$522$	0	0
$523$	−17.3586	−0.759036	−0.379518	−	0.925184i	$-0.623910\pi$
$523$	−17.3586	−0.759036	−0.379518	+	0.925184i	$0.623910\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.51052	−0.0657993
$528$	0	0
$529$	1.87636	0.0815807
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.45372	0.322856
$534$	0	0
$535$	−3.19639	−0.138192
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.641448	0.0276291
$540$	0	0
$541$	9.05446	0.389282	0.194641	−	0.980875i	$-0.437646\pi$
$541$	9.05446	0.389282	0.194641	+	0.980875i	$0.437646\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−21.3360	−0.913933
$546$	0	0
$547$	−36.8058	−1.57370	−0.786851	−	0.617143i	$-0.788290\pi$
$547$	−36.8058	−1.57370	−0.786851	+	0.617143i	$0.788290\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.98762	0.425487
$552$	0	0
$553$	3.05090	0.129737
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.34108	0.311051	0.155526	−	0.987832i	$-0.450293\pi$
$557$	7.34108	0.311051	0.155526	+	0.987832i	$0.450293\pi$
$558$	0	0
$559$	4.43268	0.187482
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.1148	−1.39562	−0.697812	−	0.716281i	$-0.745843\pi$
$563$	−33.1148	−1.39562	−0.697812	+	0.716281i	$0.745843\pi$
$564$	0	0
$565$	−15.3338	−0.645097
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.0655	−0.841188	−0.420594	−	0.907249i	$-0.638178\pi$
$569$	−20.0655	−0.841188	−0.420594	+	0.907249i	$0.638178\pi$
$570$	0	0
$571$	42.3053	1.77042	0.885211	−	0.465189i	$-0.154014\pi$
$571$	42.3053	1.77042	0.885211	+	0.465189i	$0.154014\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.3548	−0.515232
$576$	0	0
$577$	30.8923	1.28606	0.643032	−	0.765840i	$-0.277676\pi$
$577$	30.8923	1.28606	0.643032	+	0.765840i	$0.277676\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.75133	0.321580
$582$	0	0
$583$	−0.300372	−0.0124401
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.63279	−0.149941	−0.0749706	−	0.997186i	$-0.523886\pi$
$587$	−3.63279	−0.149941	−0.0749706	+	0.997186i	$0.523886\pi$
$588$	0	0
$589$	1.51052	0.0622399
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.3955	−0.878609	−0.439305	−	0.898338i	$-0.644775\pi$
$593$	−21.3955	−0.878609	−0.439305	+	0.898338i	$0.644775\pi$
$594$	0	0
$595$	1.76509	0.0723617
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.28661	−0.216005	−0.108003	−	0.994151i	$-0.534445\pi$
$599$	−5.28661	−0.216005	−0.108003	+	0.994151i	$0.534445\pi$
$600$	0	0
$601$	−20.0407	−0.817477	−0.408739	−	0.912651i	$-0.634031\pi$
$601$	−20.0407	−0.817477	−0.408739	+	0.912651i	$0.634031\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17.4523	−0.709539
$606$	0	0
$607$	8.85669	0.359482	0.179741	−	0.983714i	$-0.442474\pi$
$607$	8.85669	0.359482	0.179741	+	0.983714i	$0.442474\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.15335	0.208482
$612$	0	0
$613$	9.80223	0.395908	0.197954	−	0.980211i	$-0.436570\pi$
$613$	9.80223	0.395908	0.197954	+	0.980211i	$0.436570\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.09751	0.285735	0.142867	−	0.989742i	$-0.454368\pi$
$617$	7.09751	0.285735	0.142867	+	0.989742i	$0.454368\pi$
$618$	0	0
$619$	−32.7417	−1.31600	−0.658000	−	0.753018i	$-0.728597\pi$
$619$	−32.7417	−1.31600	−0.658000	+	0.753018i	$0.728597\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.9011	0.637065
$624$	0	0
$625$	−6.47848	−0.259139
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.69963	−0.147514
$630$	0	0
$631$	−6.86260	−0.273196	−0.136598	−	0.990627i	$-0.543617\pi$
$631$	−6.86260	−0.273196	−0.136598	+	0.990627i	$0.543617\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.9171	0.631649
$636$	0	0
$637$	5.12364	0.203006
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.80951	0.229462	0.114731	−	0.993397i	$-0.463399\pi$
$641$	5.80951	0.229462	0.114731	+	0.993397i	$0.463399\pi$
$642$	0	0
$643$	1.05680	0.0416762	0.0208381	−	0.999783i	$-0.493367\pi$
$643$	1.05680	0.0416762	0.0208381	+	0.999783i	$0.493367\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−44.0146	−1.73039	−0.865196	−	0.501435i	$-0.832806\pi$
$647$	−44.0146	−1.73039	−0.865196	+	0.501435i	$0.832806\pi$
$648$	0	0
$649$	1.36584	0.0536137
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.4917	−1.27150	−0.635749	−	0.771896i	$-0.719308\pi$
$653$	−32.4917	−1.27150	−0.635749	+	0.771896i	$0.719308\pi$
$654$	0	0
$655$	−16.8494	−0.658361
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.33751	0.0910565	0.0455283	−	0.998963i	$-0.485503\pi$
$659$	2.33751	0.0910565	0.0455283	+	0.998963i	$0.485503\pi$
$660$	0	0
$661$	1.42402	0.0553878	0.0276939	−	0.999616i	$-0.491184\pi$
$661$	1.42402	0.0553878	0.0276939	+	0.999616i	$0.491184\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.76509	−0.0684473
$666$	0	0
$667$	−49.8145	−1.92882
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.922156	0.0355995
$672$	0	0
$673$	9.58327	0.369408	0.184704	−	0.982794i	$-0.440867\pi$
$673$	9.58327	0.369408	0.184704	+	0.982794i	$0.440867\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.24219	0.0861744	0.0430872	−	0.999071i	$-0.486281\pi$
$677$	2.24219	0.0861744	0.0430872	+	0.999071i	$0.486281\pi$
$678$	0	0
$679$	16.7513	0.642857
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−40.5512	−1.55165	−0.775825	−	0.630948i	$-0.782666\pi$
$683$	−40.5512	−1.55165	−0.775825	+	0.630948i	$0.782666\pi$
$684$	0	0
$685$	−6.85022	−0.261733
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.39926	−0.0914044
$690$	0	0
$691$	−7.13093	−0.271273	−0.135637	−	0.990759i	$-0.543308\pi$
$691$	−7.13093	−0.271273	−0.135637	+	0.990759i	$0.543308\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.62769	0.0617418
$696$	0	0
$697$	−8.38688	−0.317676
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.7948	1.38972	0.694860	−	0.719145i	$-0.255466\pi$
$701$	36.7948	1.38972	0.694860	+	0.719145i	$0.255466\pi$
$702$	0	0
$703$	3.69963	0.139534
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.45744	−0.355684
$708$	0	0
$709$	−41.0690	−1.54238	−0.771190	−	0.636605i	$-0.780338\pi$
$709$	−41.0690	−1.54238	−0.771190	+	0.636605i	$0.780338\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.53390	−0.282147
$714$	0	0
$715$	0.157064	0.00587388
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.63416	0.135531	0.0677657	−	0.997701i	$-0.478413\pi$
$719$	3.63416	0.135531	0.0677657	+	0.997701i	$0.478413\pi$
$720$	0	0
$721$	9.29665	0.346225
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.7403	0.918833
$726$	0	0
$727$	12.9912	0.481816	0.240908	−	0.970548i	$-0.422555\pi$
$727$	12.9912	0.481816	0.240908	+	0.970548i	$0.422555\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.98762	−0.184474
$732$	0	0
$733$	35.4895	1.31083	0.655417	−	0.755267i	$-0.272493\pi$
$733$	35.4895	1.31083	0.655417	+	0.755267i	$0.272493\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.41026	−0.0519475
$738$	0	0
$739$	−9.26186	−0.340703	−0.170351	−	0.985383i	$-0.554490\pi$
$739$	−9.26186	−0.340703	−0.170351	+	0.985383i	$0.554490\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.0865	0.920335	0.460167	−	0.887832i	$-0.347789\pi$
$743$	25.0865	0.920335	0.460167	+	0.887832i	$0.347789\pi$
$744$	0	0
$745$	4.58836	0.168105
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.23629	−0.0817121
$750$	0	0
$751$	−30.7628	−1.12255	−0.561274	−	0.827630i	$-0.689689\pi$
$751$	−30.7628	−1.12255	−0.561274	+	0.827630i	$0.689689\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.43777	−0.234295
$756$	0	0
$757$	−45.5782	−1.65657	−0.828283	−	0.560309i	$-0.810682\pi$
$757$	−45.5782	−1.65657	−0.828283	+	0.560309i	$0.810682\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.4276	0.776749	0.388375	−	0.921502i	$-0.373037\pi$
$761$	21.4276	0.776749	0.388375	+	0.921502i	$0.373037\pi$
$762$	0	0
$763$	−14.9273	−0.540403
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.9098	0.393929
$768$	0	0
$769$	51.5178	1.85778	0.928890	−	0.370356i	$-0.120764\pi$
$769$	51.5178	1.85778	0.928890	+	0.370356i	$0.120764\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	36.5846	1.31586	0.657929	−	0.753080i	$-0.271433\pi$
$773$	36.5846	1.31586	0.657929	+	0.753080i	$0.271433\pi$
$774$	0	0
$775$	3.74171	0.134406
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.38688	0.300491
$780$	0	0
$781$	0.482196	0.0172543
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.3017	−0.688909
$786$	0	0
$787$	−11.3301	−0.403874	−0.201937	−	0.979399i	$-0.564724\pi$
$787$	−11.3301	−0.403874	−0.201937	+	0.979399i	$0.564724\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.7280	−0.381442
$792$	0	0
$793$	7.36584	0.261568
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.7242	0.804934	0.402467	−	0.915435i	$-0.368153\pi$
$797$	22.7242	0.804934	0.402467	+	0.915435i	$0.368153\pi$
$798$	0	0
$799$	−5.79851	−0.205137
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.400634	0.0141381
$804$	0	0
$805$	8.80361	0.310286
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.0580	−1.33805	−0.669024	−	0.743240i	$-0.733288\pi$
$809$	−38.0580	−1.33805	−0.669024	+	0.743240i	$0.733288\pi$
$810$	0	0
$811$	−7.16435	−0.251574	−0.125787	−	0.992057i	$-0.540146\pi$
$811$	−7.16435	−0.251574	−0.125787	+	0.992057i	$0.540146\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.05090	0.106868
$816$	0	0
$817$	4.98762	0.174495
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.5498	−0.472893	−0.236446	−	0.971645i	$-0.575983\pi$
$821$	−13.5498	−0.472893	−0.236446	+	0.971645i	$0.575983\pi$
$822$	0	0
$823$	−27.1382	−0.945979	−0.472989	−	0.881068i	$-0.656825\pi$
$823$	−27.1382	−0.945979	−0.472989	+	0.881068i	$0.656825\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−42.9308	−1.49285	−0.746425	−	0.665469i	$-0.768232\pi$
$827$	−42.9308	−1.49285	−0.746425	+	0.665469i	$0.768232\pi$
$828$	0	0
$829$	13.4981	0.468810	0.234405	−	0.972139i	$-0.424686\pi$
$829$	13.4981	0.468810	0.234405	+	0.972139i	$0.424686\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.76509	−0.199749
$834$	0	0
$835$	−2.34617	−0.0811926
$836$	0	0
$837$	0	0
$838$	0	0
$839$	16.1891	0.558910	0.279455	−	0.960159i	$-0.409846\pi$
$839$	16.1891	0.558910	0.279455	+	0.960159i	$0.409846\pi$
$840$	0	0
$841$	70.7526	2.43974
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−19.3942	−0.667179
$846$	0	0
$847$	−12.2101	−0.419546
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18.4523	−0.632538
$852$	0	0
$853$	15.9701	0.546807	0.273404	−	0.961899i	$-0.411851\pi$
$853$	15.9701	0.546807	0.273404	+	0.961899i	$0.411851\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.9070	0.475055	0.237527	−	0.971381i	$-0.423663\pi$
$857$	13.9070	0.475055	0.237527	+	0.971381i	$0.423663\pi$
$858$	0	0
$859$	17.8123	0.607747	0.303873	−	0.952712i	$-0.401720\pi$
$859$	17.8123	0.607747	0.303873	+	0.952712i	$0.401720\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.42402	0.218676	0.109338	−	0.994005i	$-0.465127\pi$
$863$	6.42402	0.218676	0.109338	+	0.994005i	$0.465127\pi$
$864$	0	0
$865$	15.0138	0.510483
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.305468	−0.0103623
$870$	0	0
$871$	−11.2646	−0.381687
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−13.1978	−0.446166
$876$	0	0
$877$	24.3745	0.823068	0.411534	−	0.911394i	$-0.364993\pi$
$877$	24.3745	0.823068	0.411534	+	0.911394i	$0.364993\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.53666	−0.220226	−0.110113	−	0.993919i	$-0.535121\pi$
$881$	−6.53666	−0.220226	−0.110113	+	0.993919i	$0.535121\pi$
$882$	0	0
$883$	−19.9913	−0.672762	−0.336381	−	0.941726i	$-0.609203\pi$
$883$	−19.9913	−0.672762	−0.336381	+	0.941726i	$0.609203\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.9208	0.635298	0.317649	−	0.948208i	$-0.397107\pi$
$887$	18.9208	0.635298	0.317649	+	0.948208i	$0.397107\pi$
$888$	0	0
$889$	11.1360	0.373490
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.79851	0.194040
$894$	0	0
$895$	−31.1978	−1.04283
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.0865	0.503163
$900$	0	0
$901$	2.69963	0.0899377
$902$	0	0
$903$	0	0
$904$	0	0
$905$	30.8777	1.02641
$906$	0	0
$907$	−5.09751	−0.169260	−0.0846300	−	0.996412i	$-0.526971\pi$
$907$	−5.09751	−0.169260	−0.0846300	+	0.996412i	$0.526971\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.25085	−0.173969	−0.0869843	−	0.996210i	$-0.527723\pi$
$911$	−5.25085	−0.173969	−0.0869843	+	0.996210i	$0.527723\pi$
$912$	0	0
$913$	−0.776094	−0.0256850
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.7883	−0.389285
$918$	0	0
$919$	39.0914	1.28951	0.644754	−	0.764390i	$-0.276960\pi$
$919$	39.0914	1.28951	0.644754	+	0.764390i	$0.276960\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.85160	0.126777
$924$	0	0
$925$	9.16435	0.301322
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.7280	0.483208	0.241604	−	0.970375i	$-0.422326\pi$
$929$	14.7280	0.483208	0.241604	+	0.970375i	$0.422326\pi$
$930$	0	0
$931$	5.76509	0.188943
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.176728	−0.00577962
$936$	0	0
$937$	−4.84294	−0.158212	−0.0791059	−	0.996866i	$-0.525207\pi$
$937$	−4.84294	−0.158212	−0.0791059	+	0.996866i	$0.525207\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.6428	−0.379545	−0.189773	−	0.981828i	$-0.560775\pi$
$941$	−11.6428	−0.379545	−0.189773	+	0.981828i	$0.560775\pi$
$942$	0	0
$943$	−41.8306	−1.36219
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.6959	1.12747	0.563733	−	0.825957i	$-0.309365\pi$
$947$	34.6959	1.12747	0.563733	+	0.825957i	$0.309365\pi$
$948$	0	0
$949$	3.20011	0.103880
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.9025	0.839064	0.419532	−	0.907741i	$-0.362194\pi$
$953$	25.9025	0.839064	0.419532	+	0.907741i	$0.362194\pi$
$954$	0	0
$955$	7.83784	0.253627
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.79261	−0.154761
$960$	0	0
$961$	−28.7183	−0.926398
$962$	0	0
$963$	0	0
$964$	0	0
$965$	23.8567	0.767974
$966$	0	0
$967$	26.9701	0.867301	0.433651	−	0.901081i	$-0.357225\pi$
$967$	26.9701	0.867301	0.433651	+	0.901081i	$0.357225\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−51.6894	−1.65879	−0.829396	−	0.558660i	$-0.811316\pi$
$971$	−51.6894	−1.65879	−0.829396	+	0.558660i	$0.811316\pi$
$972$	0	0
$973$	1.13878	0.0365076
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.09751	0.0351124	0.0175562	−	0.999846i	$-0.494411\pi$
$977$	1.09751	0.0351124	0.0175562	+	0.999846i	$0.494411\pi$
$978$	0	0
$979$	−1.59208	−0.0508832
$980$	0	0
$981$	0	0
$982$	0	0
$983$	57.3140	1.82803	0.914016	−	0.405678i	$-0.132965\pi$
$983$	57.3140	1.82803	0.914016	+	0.405678i	$0.132965\pi$
$984$	0	0
$985$	−20.3818	−0.649418
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.8764	−0.791022
$990$	0	0
$991$	−36.8850	−1.17169	−0.585846	−	0.810423i	$-0.699237\pi$
$991$	−36.8850	−1.17169	−0.585846	+	0.810423i	$0.699237\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.81818	0.152746
$996$	0	0
$997$	−40.8072	−1.29238	−0.646188	−	0.763178i	$-0.723638\pi$
$997$	−40.8072	−1.29238	−0.646188	+	0.763178i	$0.723638\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.bj.1.3		3
3.2	odd	2		8208.2.a.bk.1.1		3
4.3	odd	2		4104.2.a.f.1.3	✓	3
12.11	even	2		4104.2.a.g.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.f.1.3	✓	3	4.3	odd	2
4104.2.a.g.1.1	yes	3	12.11	even	2
8208.2.a.bj.1.3		3	1.1	even	1	trivial
8208.2.a.bk.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)

\(f(q)\)	\(=\)	\(q+1.58836 q^{5} +1.11126 q^{7} -0.111264 q^{11} -0.888736 q^{13} +1.00000 q^{17} -1.00000 q^{19} +4.98762 q^{23} -2.47710 q^{25} -9.98762 q^{29} -1.51052 q^{31} +1.76509 q^{35} -3.69963 q^{37} -8.38688 q^{41} -4.98762 q^{43} -5.79851 q^{47} -5.76509 q^{49} +2.69963 q^{53} -0.176728 q^{55} -12.2756 q^{59} -8.28799 q^{61} -1.41164 q^{65} +12.6749 q^{67} -4.33379 q^{71} -3.60074 q^{73} -0.123644 q^{77} +2.74543 q^{79} +6.97524 q^{83} +1.58836 q^{85} +14.3090 q^{89} -0.987620 q^{91} -1.58836 q^{95} +15.0741 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{5} + 3 q^{7} - 3 q^{13} + 3 q^{17} - 3 q^{19} - 3 q^{23} - 2 q^{25} - 12 q^{29} + 8 q^{31} - 12 q^{35} - 5 q^{37} + 5 q^{41} + 3 q^{43} + 7 q^{47} + 2 q^{53} + 11 q^{55} - 7 q^{59} - 13 q^{61}+ \cdots - 9 q^{97}+O(q^{100}) \) 3 * q - q^5 + 3 * q^7 - 3 * q^13 + 3 * q^17 - 3 * q^19 - 3 * q^23 - 2 * q^25 - 12 * q^29 + 8 * q^31 - 12 * q^35 - 5 * q^37 + 5 * q^41 + 3 * q^43 + 7 * q^47 + 2 * q^53 + 11 * q^55 - 7 * q^59 - 13 * q^61 - 10 * q^65 - 4 * q^67 - 12 * q^71 - 23 * q^73 - 18 * q^77 + 13 * q^79 - 15 * q^83 - q^85 + 6 * q^89 + 15 * q^91 + q^95 - 9 * q^97

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