LMFDB - Embedded newform 2592.2.a.u.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(1,2592)] 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("2592.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-2,0,0,0,0,0,0,0,6,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13068.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 1 $ x^4 - x^3 - 6*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.04374$ of defining polynomial
Character	$\chi$	$=$	2592.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37228 q^{5} +2.20979 q^{7} -5.93580 q^{11} +4.37228 q^{13} +3.37228 q^{17} -3.72601 q^{19} +2.20979 q^{23} +0.627719 q^{25} +0.372281 q^{29} +9.66181 q^{31} +5.24224 q^{35} +4.00000 q^{37} +1.00000 q^{41} +5.93580 q^{43} +2.20979 q^{47} -2.11684 q^{49} -4.00000 q^{53} -14.0814 q^{55} +10.3554 q^{59} +15.1168 q^{61} +10.3723 q^{65} -10.3554 q^{67} +4.41957 q^{71} +4.62772 q^{73} -13.1168 q^{77} -9.66181 q^{79} -14.0814 q^{83} +8.00000 q^{85} +1.25544 q^{89} +9.66181 q^{91} -8.83915 q^{95} +9.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 6 q^{13} + 2 q^{17} + 14 q^{25} - 10 q^{29} + 16 q^{37} + 4 q^{41} + 26 q^{49} - 16 q^{53} + 26 q^{61} + 30 q^{65} + 30 q^{73} - 18 q^{77} + 32 q^{85} + 28 q^{89} + 36 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 6 * q^13 + 2 * q^17 + 14 * q^25 - 10 * q^29 + 16 * q^37 + 4 * q^41 + 26 * q^49 - 16 * q^53 + 26 * q^61 + 30 * q^65 + 30 * q^73 - 18 * q^77 + 32 * q^85 + 28 * q^89 + 36 * 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
$5$	2.37228	1.06092	0.530458	−	0.847711i	$-0.322020\pi$
$5$	2.37228	1.06092	0.530458	+	0.847711i	$0.322020\pi$
$6$	0	0
$7$	2.20979	0.835221	0.417610	−	0.908626i	$-0.362868\pi$
$7$	2.20979	0.835221	0.417610	+	0.908626i	$0.362868\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.93580	−1.78971	−0.894855	−	0.446357i	$-0.852721\pi$
$11$	−5.93580	−1.78971	−0.894855	+	0.446357i	$0.852721\pi$
$12$	0	0
$13$	4.37228	1.21265	0.606326	−	0.795216i	$-0.292643\pi$
$13$	4.37228	1.21265	0.606326	+	0.795216i	$0.292643\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.37228	0.817898	0.408949	−	0.912557i	$-0.365895\pi$
$17$	3.37228	0.817898	0.408949	+	0.912557i	$0.365895\pi$
$18$	0	0
$19$	−3.72601	−0.854805	−0.427403	−	0.904061i	$-0.640571\pi$
$19$	−3.72601	−0.854805	−0.427403	+	0.904061i	$0.640571\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.20979	0.460772	0.230386	−	0.973099i	$-0.426001\pi$
$23$	2.20979	0.460772	0.230386	+	0.973099i	$0.426001\pi$
$24$	0	0
$25$	0.627719	0.125544
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.372281	0.0691309	0.0345655	−	0.999402i	$-0.488995\pi$
$29$	0.372281	0.0691309	0.0345655	+	0.999402i	$0.488995\pi$
$30$	0	0
$31$	9.66181	1.73531	0.867656	−	0.497165i	$-0.165626\pi$
$31$	9.66181	1.73531	0.867656	+	0.497165i	$0.165626\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.24224	0.886099
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	5.93580	0.905201	0.452600	−	0.891714i	$-0.350496\pi$
$43$	5.93580	0.905201	0.452600	+	0.891714i	$0.350496\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.20979	0.322330	0.161165	−	0.986927i	$-0.448475\pi$
$47$	2.20979	0.322330	0.161165	+	0.986927i	$0.448475\pi$
$48$	0	0
$49$	−2.11684	−0.302406
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−14.0814	−1.89873
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3554	1.34815	0.674077	−	0.738661i	$-0.264542\pi$
$59$	10.3554	1.34815	0.674077	+	0.738661i	$0.264542\pi$
$60$	0	0
$61$	15.1168	1.93551	0.967757	−	0.251887i	$-0.0810510\pi$
$61$	15.1168	1.93551	0.967757	+	0.251887i	$0.0810510\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.3723	1.28652
$66$	0	0
$67$	−10.3554	−1.26511	−0.632555	−	0.774516i	$-0.717994\pi$
$67$	−10.3554	−1.26511	−0.632555	+	0.774516i	$0.717994\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.41957	0.524507	0.262253	−	0.964999i	$-0.415534\pi$
$71$	4.41957	0.524507	0.262253	+	0.964999i	$0.415534\pi$
$72$	0	0
$73$	4.62772	0.541634	0.270817	−	0.962631i	$-0.412706\pi$
$73$	4.62772	0.541634	0.270817	+	0.962631i	$0.412706\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.1168	−1.49480
$78$	0	0
$79$	−9.66181	−1.08704	−0.543519	−	0.839397i	$-0.682908\pi$
$79$	−9.66181	−1.08704	−0.543519	+	0.839397i	$0.682908\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.0814	−1.54563	−0.772816	−	0.634630i	$-0.781153\pi$
$83$	−14.0814	−1.54563	−0.772816	+	0.634630i	$0.781153\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.25544	0.133076	0.0665380	−	0.997784i	$-0.478805\pi$
$89$	1.25544	0.133076	0.0665380	+	0.997784i	$0.478805\pi$
$90$	0	0
$91$	9.66181	1.01283
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.83915	−0.906877
$96$	0	0
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.1168	−1.50418	−0.752091	−	0.659059i	$-0.770955\pi$
$101$	−15.1168	−1.50418	−0.752091	+	0.659059i	$0.770955\pi$
$102$	0	0
$103$	−2.20979	−0.217737	−0.108868	−	0.994056i	$-0.534723\pi$
$103$	−2.20979	−0.217737	−0.108868	+	0.994056i	$0.534723\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.72601	−0.360207	−0.180104	−	0.983648i	$-0.557643\pi$
$107$	−3.72601	−0.360207	−0.180104	+	0.983648i	$0.557643\pi$
$108$	0	0
$109$	17.4891	1.67515	0.837577	−	0.546319i	$-0.183971\pi$
$109$	17.4891	1.67515	0.837577	+	0.546319i	$0.183971\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.6277	1.09384	0.546922	−	0.837184i	$-0.315799\pi$
$113$	11.6277	1.09384	0.546922	+	0.837184i	$0.315799\pi$
$114$	0	0
$115$	5.24224	0.488841
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.45202	0.683126
$120$	0	0
$121$	24.2337	2.20306
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.3723	−0.927725
$126$	0	0
$127$	7.45202	0.661260	0.330630	−	0.943760i	$-0.392739\pi$
$127$	7.45202	0.661260	0.330630	+	0.943760i	$0.392739\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−9.66181	−0.844156	−0.422078	−	0.906560i	$-0.638699\pi$
$131$	−9.66181	−0.844156	−0.422078	+	0.906560i	$0.638699\pi$
$132$	0	0
$133$	−8.23369	−0.713951
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.4891	−1.06702	−0.533509	−	0.845794i	$-0.679127\pi$
$137$	−12.4891	−1.06702	−0.533509	+	0.845794i	$0.679127\pi$
$138$	0	0
$139$	2.90335	0.246259	0.123129	−	0.992391i	$-0.460707\pi$
$139$	2.90335	0.246259	0.123129	+	0.992391i	$0.460707\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−25.9530	−2.17030
$144$	0	0
$145$	0.883156	0.0733421
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.6277	−0.952580	−0.476290	−	0.879288i	$-0.658019\pi$
$149$	−11.6277	−0.952580	−0.476290	+	0.879288i	$0.658019\pi$
$150$	0	0
$151$	−2.20979	−0.179830	−0.0899149	−	0.995949i	$-0.528660\pi$
$151$	−2.20979	−0.179830	−0.0899149	+	0.995949i	$0.528660\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	22.9205	1.84102
$156$	0	0
$157$	−3.11684	−0.248751	−0.124376	−	0.992235i	$-0.539693\pi$
$157$	−3.11684	−0.248751	−0.124376	+	0.992235i	$0.539693\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.88316	0.384847
$162$	0	0
$163$	8.83915	0.692335	0.346168	−	0.938173i	$-0.387483\pi$
$163$	8.83915	0.692335	0.346168	+	0.938173i	$0.387483\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.66181	−0.747653	−0.373827	−	0.927499i	$-0.621954\pi$
$167$	−9.66181	−0.747653	−0.373827	+	0.927499i	$0.621954\pi$
$168$	0	0
$169$	6.11684	0.470526
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.372281	−0.0283040	−0.0141520	−	0.999900i	$-0.504505\pi$
$173$	−0.372281	−0.0283040	−0.0141520	+	0.999900i	$0.504505\pi$
$174$	0	0
$175$	1.38712	0.104857
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.83915	0.660669	0.330334	−	0.943864i	$-0.392838\pi$
$179$	8.83915	0.660669	0.330334	+	0.943864i	$0.392838\pi$
$180$	0	0
$181$	−8.23369	−0.612005	−0.306003	−	0.952031i	$-0.598992\pi$
$181$	−8.23369	−0.612005	−0.306003	+	0.952031i	$0.598992\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.48913	0.697654
$186$	0	0
$187$	−20.0172	−1.46380
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.1138	1.23831	0.619157	−	0.785268i	$-0.287475\pi$
$191$	17.1138	1.23831	0.619157	+	0.785268i	$0.287475\pi$
$192$	0	0
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.744563	0.0530479	0.0265239	−	0.999648i	$-0.491556\pi$
$197$	0.744563	0.0530479	0.0265239	+	0.999648i	$0.491556\pi$
$198$	0	0
$199$	20.7107	1.46815	0.734073	−	0.679071i	$-0.237617\pi$
$199$	20.7107	1.46815	0.734073	+	0.679071i	$0.237617\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.822662	0.0577396
$204$	0	0
$205$	2.37228	0.165687
$206$	0	0
$207$	0	0
$208$	0	0
$209$	22.1168	1.52985
$210$	0	0
$211$	−5.24224	−0.360890	−0.180445	−	0.983585i	$-0.557754\pi$
$211$	−5.24224	−0.360890	−0.180445	+	0.983585i	$0.557754\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.0814	0.960342
$216$	0	0
$217$	21.3505	1.44937
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.7446	0.991827
$222$	0	0
$223$	2.20979	0.147978	0.0739891	−	0.997259i	$-0.476427\pi$
$223$	2.20979	0.147978	0.0739891	+	0.997259i	$0.476427\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.96825	0.595243	0.297622	−	0.954684i	$-0.403807\pi$
$227$	8.96825	0.595243	0.297622	+	0.954684i	$0.403807\pi$
$228$	0	0
$229$	−4.37228	−0.288928	−0.144464	−	0.989510i	$-0.546146\pi$
$229$	−4.37228	−0.288928	−0.144464	+	0.989510i	$0.546146\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.86141	0.187457	0.0937285	−	0.995598i	$-0.470121\pi$
$233$	2.86141	0.187457	0.0937285	+	0.995598i	$0.470121\pi$
$234$	0	0
$235$	5.24224	0.341966
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.1138	1.10700	0.553501	−	0.832849i	$-0.313292\pi$
$239$	17.1138	1.10700	0.553501	+	0.832849i	$0.313292\pi$
$240$	0	0
$241$	0.255437	0.0164542	0.00822708	−	0.999966i	$-0.497381\pi$
$241$	0.255437	0.0164542	0.00822708	+	0.999966i	$0.497381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.02175	−0.320828
$246$	0	0
$247$	−16.2912	−1.03658
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.1780	−0.705551	−0.352776	−	0.935708i	$-0.614762\pi$
$251$	−11.1780	−0.705551	−0.352776	+	0.935708i	$0.614762\pi$
$252$	0	0
$253$	−13.1168	−0.824649
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.48913	−0.404781	−0.202390	−	0.979305i	$-0.564871\pi$
$257$	−6.48913	−0.404781	−0.202390	+	0.979305i	$0.564871\pi$
$258$	0	0
$259$	8.83915	0.549238
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.1138	−1.05528	−0.527642	−	0.849467i	$-0.676924\pi$
$263$	−17.1138	−1.05528	−0.527642	+	0.849467i	$0.676924\pi$
$264$	0	0
$265$	−9.48913	−0.582912
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.48913	0.578562	0.289281	−	0.957244i	$-0.406584\pi$
$269$	9.48913	0.578562	0.289281	+	0.957244i	$0.406584\pi$
$270$	0	0
$271$	14.9040	0.905356	0.452678	−	0.891674i	$-0.350469\pi$
$271$	14.9040	0.905356	0.452678	+	0.891674i	$0.350469\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.72601	−0.224687
$276$	0	0
$277$	13.1168	0.788115	0.394057	−	0.919086i	$-0.371071\pi$
$277$	13.1168	0.788115	0.394057	+	0.919086i	$0.371071\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	25.8614	1.54276	0.771381	−	0.636373i	$-0.219566\pi$
$281$	25.8614	1.54276	0.771381	+	0.636373i	$0.219566\pi$
$282$	0	0
$283$	5.24224	0.311619	0.155809	−	0.987787i	$-0.450201\pi$
$283$	5.24224	0.311619	0.155809	+	0.987787i	$0.450201\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.20979	0.130440
$288$	0	0
$289$	−5.62772	−0.331042
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.3723	−1.19016	−0.595081	−	0.803666i	$-0.702880\pi$
$293$	−20.3723	−1.19016	−0.595081	+	0.803666i	$0.702880\pi$
$294$	0	0
$295$	24.5659	1.43028
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.66181	0.558757
$300$	0	0
$301$	13.1168	0.756042
$302$	0	0
$303$	0	0
$304$	0	0
$305$	35.8614	2.05342
$306$	0	0
$307$	−28.8563	−1.64692	−0.823459	−	0.567376i	$-0.807959\pi$
$307$	−28.8563	−1.64692	−0.823459	+	0.567376i	$0.807959\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.6943	−0.719825	−0.359913	−	0.932986i	$-0.617194\pi$
$311$	−12.6943	−0.719825	−0.359913	+	0.932986i	$0.617194\pi$
$312$	0	0
$313$	19.2337	1.08715	0.543576	−	0.839360i	$-0.317070\pi$
$313$	19.2337	1.08715	0.543576	+	0.839360i	$0.317070\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.37228	0.470234	0.235117	−	0.971967i	$-0.424453\pi$
$317$	8.37228	0.470234	0.235117	+	0.971967i	$0.424453\pi$
$318$	0	0
$319$	−2.20979	−0.123724
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12.5652	−0.699144
$324$	0	0
$325$	2.74456	0.152241
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.88316	0.269217
$330$	0	0
$331$	−33.4050	−1.83610	−0.918052	−	0.396459i	$-0.870239\pi$
$331$	−33.4050	−1.83610	−0.918052	+	0.396459i	$0.870239\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−24.5659	−1.34218
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−57.3505	−3.10571
$342$	0	0
$343$	−20.1463	−1.08780
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.3554	−0.555905	−0.277953	−	0.960595i	$-0.589656\pi$
$347$	−10.3554	−0.555905	−0.277953	+	0.960595i	$0.589656\pi$
$348$	0	0
$349$	−23.1168	−1.23742	−0.618708	−	0.785621i	$-0.712344\pi$
$349$	−23.1168	−1.23742	−0.618708	+	0.785621i	$0.712344\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−19.2337	−1.02371	−0.511853	−	0.859073i	$-0.671041\pi$
$353$	−19.2337	−1.02371	−0.511853	+	0.859073i	$0.671041\pi$
$354$	0	0
$355$	10.4845	0.556458
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.3561	1.17991	0.589954	−	0.807437i	$-0.299146\pi$
$359$	22.3561	1.17991	0.589954	+	0.807437i	$0.299146\pi$
$360$	0	0
$361$	−5.11684	−0.269308
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.9783	0.574628
$366$	0	0
$367$	−27.3401	−1.42714	−0.713571	−	0.700583i	$-0.752923\pi$
$367$	−27.3401	−1.42714	−0.713571	+	0.700583i	$0.752923\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.83915	−0.458906
$372$	0	0
$373$	−17.1168	−0.886277	−0.443138	−	0.896453i	$-0.646135\pi$
$373$	−17.1168	−0.886277	−0.443138	+	0.896453i	$0.646135\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.62772	0.0838318
$378$	0	0
$379$	−8.14558	−0.418411	−0.209205	−	0.977872i	$-0.567088\pi$
$379$	−8.14558	−0.418411	−0.209205	+	0.977872i	$0.567088\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.9205	1.17118	0.585592	−	0.810606i	$-0.300862\pi$
$383$	22.9205	1.17118	0.585592	+	0.810606i	$0.300862\pi$
$384$	0	0
$385$	−31.1168	−1.58586
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−19.3505	−0.981111	−0.490555	−	0.871410i	$-0.663206\pi$
$389$	−19.3505	−0.981111	−0.490555	+	0.871410i	$0.663206\pi$
$390$	0	0
$391$	7.45202	0.376865
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.9205	−1.15326
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.9783	1.39717	0.698584	−	0.715528i	$-0.253814\pi$
$401$	27.9783	1.39717	0.698584	+	0.715528i	$0.253814\pi$
$402$	0	0
$403$	42.2441	2.10433
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.7432	−1.17691
$408$	0	0
$409$	−18.2554	−0.902673	−0.451337	−	0.892354i	$-0.649053\pi$
$409$	−18.2554	−0.902673	−0.451337	+	0.892354i	$0.649053\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	22.8832	1.12601
$414$	0	0
$415$	−33.4050	−1.63979
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5.24224	−0.256100	−0.128050	−	0.991768i	$-0.540872\pi$
$419$	−5.24224	−0.256100	−0.128050	+	0.991768i	$0.540872\pi$
$420$	0	0
$421$	−22.3723	−1.09036	−0.545179	−	0.838320i	$-0.683538\pi$
$421$	−22.3723	−1.09036	−0.545179	+	0.838320i	$0.683538\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.11684	0.102682
$426$	0	0
$427$	33.4050	1.61658
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.45202	0.358951	0.179476	−	0.983762i	$-0.442560\pi$
$431$	7.45202	0.358951	0.179476	+	0.983762i	$0.442560\pi$
$432$	0	0
$433$	−18.1168	−0.870640	−0.435320	−	0.900276i	$-0.643365\pi$
$433$	−18.1168	−0.870640	−0.435320	+	0.900276i	$0.643365\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.23369	−0.393871
$438$	0	0
$439$	22.9205	1.09394	0.546969	−	0.837153i	$-0.315782\pi$
$439$	22.9205	1.09394	0.546969	+	0.837153i	$0.315782\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.96825	−0.426094	−0.213047	−	0.977042i	$-0.568339\pi$
$443$	−8.96825	−0.426094	−0.213047	+	0.977042i	$0.568339\pi$
$444$	0	0
$445$	2.97825	0.141183
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.1168	−0.854987	−0.427493	−	0.904018i	$-0.640603\pi$
$449$	−18.1168	−0.854987	−0.427493	+	0.904018i	$0.640603\pi$
$450$	0	0
$451$	−5.93580	−0.279506
$452$	0	0
$453$	0	0
$454$	0	0
$455$	22.9205	1.07453
$456$	0	0
$457$	−25.9783	−1.21521	−0.607606	−	0.794239i	$-0.707870\pi$
$457$	−25.9783	−1.21521	−0.607606	+	0.794239i	$0.707870\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.1168	−1.35611	−0.678053	−	0.735013i	$-0.737176\pi$
$461$	−29.1168	−1.35611	−0.678053	+	0.735013i	$0.737176\pi$
$462$	0	0
$463$	−37.8246	−1.75786	−0.878928	−	0.476954i	$-0.841741\pi$
$463$	−37.8246	−1.75786	−0.878928	+	0.476954i	$0.841741\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.3083	1.68015	0.840075	−	0.542470i	$-0.182511\pi$
$467$	36.3083	1.68015	0.840075	+	0.542470i	$0.182511\pi$
$468$	0	0
$469$	−22.8832	−1.05665
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−35.2337	−1.62005
$474$	0	0
$475$	−2.33889	−0.107315
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.20979	−0.100968	−0.0504839	−	0.998725i	$-0.516076\pi$
$479$	−2.20979	−0.100968	−0.0504839	+	0.998725i	$0.516076\pi$
$480$	0	0
$481$	17.4891	0.797435
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21.3505	0.969478
$486$	0	0
$487$	7.45202	0.337683	0.168842	−	0.985643i	$-0.445997\pi$
$487$	7.45202	0.337683	0.168842	+	0.985643i	$0.445997\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.129100	0.00582621	0.00291310	−	0.999996i	$-0.499073\pi$
$491$	0.129100	0.00582621	0.00291310	+	0.999996i	$0.499073\pi$
$492$	0	0
$493$	1.25544	0.0565421
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.76631	0.438079
$498$	0	0
$499$	25.2594	1.13077	0.565383	−	0.824828i	$-0.308728\pi$
$499$	25.2594	1.13077	0.565383	+	0.824828i	$0.308728\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10.4845	0.467479	0.233740	−	0.972299i	$-0.424904\pi$
$503$	10.4845	0.467479	0.233740	+	0.972299i	$0.424904\pi$
$504$	0	0
$505$	−35.8614	−1.59581
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.3505	−0.591752	−0.295876	−	0.955226i	$-0.595612\pi$
$509$	−13.3505	−0.591752	−0.295876	+	0.955226i	$0.595612\pi$
$510$	0	0
$511$	10.2263	0.452384
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.24224	−0.231000
$516$	0	0
$517$	−13.1168	−0.576878
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.3505	−0.716330	−0.358165	−	0.933658i	$-0.616597\pi$
$521$	−16.3505	−0.716330	−0.358165	+	0.933658i	$0.616597\pi$
$522$	0	0
$523$	−19.3236	−0.844963	−0.422481	−	0.906372i	$-0.638841\pi$
$523$	−19.3236	−0.844963	−0.422481	+	0.906372i	$0.638841\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.5823	1.41931
$528$	0	0
$529$	−18.1168	−0.787689
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.37228	0.189385
$534$	0	0
$535$	−8.83915	−0.382150
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.5652	0.541220
$540$	0	0
$541$	8.74456	0.375958	0.187979	−	0.982173i	$-0.439806\pi$
$541$	8.74456	0.375958	0.187979	+	0.982173i	$0.439806\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	41.4891	1.77720
$546$	0	0
$547$	1.51622	0.0648291	0.0324145	−	0.999475i	$-0.489680\pi$
$547$	1.51622	0.0648291	0.0324145	+	0.999475i	$0.489680\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.38712	−0.0590935
$552$	0	0
$553$	−21.3505	−0.907917
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.7446	−0.794233	−0.397116	−	0.917768i	$-0.629989\pi$
$557$	−18.7446	−0.794233	−0.397116	+	0.917768i	$0.629989\pi$
$558$	0	0
$559$	25.9530	1.09769
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.2270	0.936755	0.468377	−	0.883528i	$-0.344839\pi$
$563$	22.2270	0.936755	0.468377	+	0.883528i	$0.344839\pi$
$564$	0	0
$565$	27.5842	1.16048
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.7446	−1.66618	−0.833089	−	0.553138i	$-0.813430\pi$
$569$	−39.7446	−1.66618	−0.833089	+	0.553138i	$0.813430\pi$
$570$	0	0
$571$	−0.129100	−0.00540267	−0.00270134	−	0.999996i	$-0.500860\pi$
$571$	−0.129100	−0.00540267	−0.00270134	+	0.999996i	$0.500860\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.38712	0.0578471
$576$	0	0
$577$	18.1168	0.754214	0.377107	−	0.926170i	$-0.376919\pi$
$577$	18.1168	0.754214	0.377107	+	0.926170i	$0.376919\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−31.1168	−1.29094
$582$	0	0
$583$	23.7432	0.983342
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.7425	−0.484665	−0.242332	−	0.970193i	$-0.577912\pi$
$587$	−11.7425	−0.484665	−0.242332	+	0.970193i	$0.577912\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−17.7228	−0.727789	−0.363894	−	0.931440i	$-0.618553\pi$
$593$	−17.7228	−0.727789	−0.363894	+	0.931440i	$0.618553\pi$
$594$	0	0
$595$	17.6783	0.724739
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.3401	−1.11709	−0.558543	−	0.829476i	$-0.688639\pi$
$599$	−27.3401	−1.11709	−0.558543	+	0.829476i	$0.688639\pi$
$600$	0	0
$601$	−2.76631	−0.112840	−0.0564201	−	0.998407i	$-0.517969\pi$
$601$	−2.76631	−0.112840	−0.0564201	+	0.998407i	$0.517969\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	57.4891	2.33727
$606$	0	0
$607$	37.8246	1.53525	0.767626	−	0.640898i	$-0.221438\pi$
$607$	37.8246	1.53525	0.767626	+	0.640898i	$0.221438\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.66181	0.390875
$612$	0	0
$613$	36.4674	1.47290	0.736452	−	0.676490i	$-0.236500\pi$
$613$	36.4674	1.47290	0.736452	+	0.676490i	$0.236500\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−37.7446	−1.51954	−0.759769	−	0.650193i	$-0.774688\pi$
$617$	−37.7446	−1.51954	−0.759769	+	0.650193i	$0.774688\pi$
$618$	0	0
$619$	1.51622	0.0609422	0.0304711	−	0.999536i	$-0.490299\pi$
$619$	1.51622	0.0609422	0.0304711	+	0.999536i	$0.490299\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.77425	0.111148
$624$	0	0
$625$	−27.7446	−1.10978
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.4891	0.537847
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17.6783	0.701542
$636$	0	0
$637$	−9.25544	−0.366714
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.7446	0.542878	0.271439	−	0.962456i	$-0.412501\pi$
$641$	13.7446	0.542878	0.271439	+	0.962456i	$0.412501\pi$
$642$	0	0
$643$	−17.8074	−0.702255	−0.351127	−	0.936328i	$-0.614202\pi$
$643$	−17.8074	−0.702255	−0.351127	+	0.936328i	$0.614202\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.0344	−1.57391	−0.786956	−	0.617009i	$-0.788344\pi$
$647$	−40.0344	−1.57391	−0.786956	+	0.617009i	$0.788344\pi$
$648$	0	0
$649$	−61.4674	−2.41281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.1168	1.60903	0.804513	−	0.593935i	$-0.202426\pi$
$653$	41.1168	1.60903	0.804513	+	0.593935i	$0.202426\pi$
$654$	0	0
$655$	−22.9205	−0.895579
$656$	0	0
$657$	0	0
$658$	0	0
$659$	31.7597	1.23718	0.618591	−	0.785713i	$-0.287704\pi$
$659$	31.7597	1.23718	0.618591	+	0.785713i	$0.287704\pi$
$660$	0	0
$661$	−3.11684	−0.121231	−0.0606156	−	0.998161i	$-0.519306\pi$
$661$	−3.11684	−0.121231	−0.0606156	+	0.998161i	$0.519306\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−19.5326	−0.757443
$666$	0	0
$667$	0.822662	0.0318536
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−89.7305	−3.46401
$672$	0	0
$673$	23.3505	0.900097	0.450048	−	0.893004i	$-0.351407\pi$
$673$	23.3505	0.900097	0.450048	+	0.893004i	$0.351407\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.35053	0.0519052	0.0259526	−	0.999663i	$-0.491738\pi$
$677$	1.35053	0.0519052	0.0259526	+	0.999663i	$0.491738\pi$
$678$	0	0
$679$	19.8881	0.763234
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−36.3083	−1.38930	−0.694650	−	0.719348i	$-0.744441\pi$
$683$	−36.3083	−1.38930	−0.694650	+	0.719348i	$0.744441\pi$
$684$	0	0
$685$	−29.6277	−1.13202
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−17.4891	−0.666283
$690$	0	0
$691$	−46.6637	−1.77517	−0.887586	−	0.460643i	$-0.847619\pi$
$691$	−46.6637	−1.77517	−0.887586	+	0.460643i	$0.847619\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.88756	0.261260
$696$	0	0
$697$	3.37228	0.127734
$698$	0	0
$699$	0	0
$700$	0	0
$701$	34.7446	1.31228	0.656142	−	0.754637i	$-0.272187\pi$
$701$	34.7446	1.31228	0.656142	+	0.754637i	$0.272187\pi$
$702$	0	0
$703$	−14.9040	−0.562117
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−33.4050	−1.25632
$708$	0	0
$709$	−12.6060	−0.473427	−0.236714	−	0.971579i	$-0.576070\pi$
$709$	−12.6060	−0.473427	−0.236714	+	0.971579i	$0.576070\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21.3505	0.799584
$714$	0	0
$715$	−61.5678	−2.30250
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.5823	−1.21512	−0.607558	−	0.794275i	$-0.707851\pi$
$719$	−32.5823	−1.21512	−0.607558	+	0.794275i	$0.707851\pi$
$720$	0	0
$721$	−4.88316	−0.181858
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.233688	0.00867895
$726$	0	0
$727$	−34.7921	−1.29037	−0.645184	−	0.764027i	$-0.723219\pi$
$727$	−34.7921	−1.29037	−0.645184	+	0.764027i	$0.723219\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	20.0172	0.740362
$732$	0	0
$733$	22.8832	0.845209	0.422604	−	0.906314i	$-0.361116\pi$
$733$	22.8832	0.845209	0.422604	+	0.906314i	$0.361116\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	61.4674	2.26418
$738$	0	0
$739$	15.5976	0.573767	0.286884	−	0.957965i	$-0.407381\pi$
$739$	15.5976	0.573767	0.286884	+	0.957965i	$0.407381\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.5983	1.01248	0.506242	−	0.862392i	$-0.331034\pi$
$743$	27.5983	1.01248	0.506242	+	0.862392i	$0.331034\pi$
$744$	0	0
$745$	−27.5842	−1.01061
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.23369	−0.300852
$750$	0	0
$751$	22.9205	0.836382	0.418191	−	0.908359i	$-0.362664\pi$
$751$	22.9205	0.836382	0.418191	+	0.908359i	$0.362664\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.24224	−0.190784
$756$	0	0
$757$	−34.4674	−1.25274	−0.626369	−	0.779527i	$-0.715460\pi$
$757$	−34.4674	−1.25274	−0.626369	+	0.779527i	$0.715460\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.3723	0.738495	0.369247	−	0.929331i	$-0.379616\pi$
$761$	20.3723	0.738495	0.369247	+	0.929331i	$0.379616\pi$
$762$	0	0
$763$	38.6472	1.39912
$764$	0	0
$765$	0	0
$766$	0	0
$767$	45.2766	1.63484
$768$	0	0
$769$	−14.8832	−0.536700	−0.268350	−	0.963321i	$-0.586478\pi$
$769$	−14.8832	−0.536700	−0.268350	+	0.963321i	$0.586478\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−18.7446	−0.674195	−0.337098	−	0.941470i	$-0.609445\pi$
$773$	−18.7446	−0.674195	−0.337098	+	0.941470i	$0.609445\pi$
$774$	0	0
$775$	6.06490	0.217858
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.72601	−0.133498
$780$	0	0
$781$	−26.2337	−0.938715
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.39403	−0.263904
$786$	0	0
$787$	−42.2441	−1.50584	−0.752921	−	0.658111i	$-0.771356\pi$
$787$	−42.2441	−1.50584	−0.752921	+	0.658111i	$0.771356\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	25.6948	0.913601
$792$	0	0
$793$	66.0951	2.34711
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.0951	1.63277	0.816386	−	0.577507i	$-0.195974\pi$
$797$	46.0951	1.63277	0.816386	+	0.577507i	$0.195974\pi$
$798$	0	0
$799$	7.45202	0.263634
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−27.4692	−0.969367
$804$	0	0
$805$	11.5842	0.408290
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.6277	−0.443967	−0.221983	−	0.975050i	$-0.571253\pi$
$809$	−12.6277	−0.443967	−0.221983	+	0.975050i	$0.571253\pi$
$810$	0	0
$811$	33.5341	1.17754	0.588771	−	0.808300i	$-0.299612\pi$
$811$	33.5341	1.17754	0.588771	+	0.808300i	$0.299612\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.9689	0.734510
$816$	0	0
$817$	−22.1168	−0.773770
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.37228	0.292195	0.146097	−	0.989270i	$-0.453329\pi$
$821$	8.37228	0.292195	0.146097	+	0.989270i	$0.453329\pi$
$822$	0	0
$823$	−22.9205	−0.798959	−0.399480	−	0.916742i	$-0.630809\pi$
$823$	−22.9205	−0.798959	−0.399480	+	0.916742i	$0.630809\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.83915	0.307367	0.153684	−	0.988120i	$-0.450886\pi$
$827$	8.83915	0.307367	0.153684	+	0.988120i	$0.450886\pi$
$828$	0	0
$829$	−8.23369	−0.285968	−0.142984	−	0.989725i	$-0.545670\pi$
$829$	−8.23369	−0.285968	−0.142984	+	0.989725i	$0.545670\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.13859	−0.247338
$834$	0	0
$835$	−22.9205	−0.793198
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.66181	0.333563	0.166781	−	0.985994i	$-0.446663\pi$
$839$	9.66181	0.333563	0.166781	+	0.985994i	$0.446663\pi$
$840$	0	0
$841$	−28.8614	−0.995221
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.5109	0.499189
$846$	0	0
$847$	53.5513	1.84004
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.83915	0.303002
$852$	0	0
$853$	−21.1168	−0.723027	−0.361513	−	0.932367i	$-0.617740\pi$
$853$	−21.1168	−0.723027	−0.361513	+	0.932367i	$0.617740\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.0951	−1.50626	−0.753130	−	0.657872i	$-0.771457\pi$
$857$	−44.0951	−1.50626	−0.753130	+	0.657872i	$0.771457\pi$
$858$	0	0
$859$	−17.8074	−0.607580	−0.303790	−	0.952739i	$-0.598252\pi$
$859$	−17.8074	−0.607580	−0.303790	+	0.952739i	$0.598252\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.6783	0.601776	0.300888	−	0.953659i	$-0.402717\pi$
$863$	17.6783	0.601776	0.300888	+	0.953659i	$0.402717\pi$
$864$	0	0
$865$	−0.883156	−0.0300282
$866$	0	0
$867$	0	0
$868$	0	0
$869$	57.3505	1.94548
$870$	0	0
$871$	−45.2766	−1.53414
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−22.9205	−0.774855
$876$	0	0
$877$	−13.3505	−0.450815	−0.225408	−	0.974265i	$-0.572371\pi$
$877$	−13.3505	−0.450815	−0.225408	+	0.974265i	$0.572371\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.23369	0.142637	0.0713183	−	0.997454i	$-0.477279\pi$
$881$	4.23369	0.142637	0.0713183	+	0.997454i	$0.477279\pi$
$882$	0	0
$883$	21.4043	0.720312	0.360156	−	0.932892i	$-0.382723\pi$
$883$	21.4043	0.720312	0.360156	+	0.932892i	$0.382723\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.9205	−0.769596	−0.384798	−	0.923001i	$-0.625729\pi$
$887$	−22.9205	−0.769596	−0.384798	+	0.923001i	$0.625729\pi$
$888$	0	0
$889$	16.4674	0.552298
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.23369	−0.275530
$894$	0	0
$895$	20.9689	0.700914
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.59691	0.119964
$900$	0	0
$901$	−13.4891	−0.449388
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.5326	−0.649286
$906$	0	0
$907$	28.2919	0.939416	0.469708	−	0.882822i	$-0.344359\pi$
$907$	28.2919	0.939416	0.469708	+	0.882822i	$0.344359\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52.7286	1.74698	0.873488	−	0.486845i	$-0.161852\pi$
$911$	52.7286	1.74698	0.873488	+	0.486845i	$0.161852\pi$
$912$	0	0
$913$	83.5842	2.76623
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.3505	−0.705057
$918$	0	0
$919$	−25.1303	−0.828973	−0.414486	−	0.910056i	$-0.636039\pi$
$919$	−25.1303	−0.828973	−0.414486	+	0.910056i	$0.636039\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	19.3236	0.636045
$924$	0	0
$925$	2.51087	0.0825571
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.3723	1.19334	0.596668	−	0.802488i	$-0.296491\pi$
$929$	36.3723	1.19334	0.596668	+	0.802488i	$0.296491\pi$
$930$	0	0
$931$	7.88738	0.258499
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−47.4864	−1.55297
$936$	0	0
$937$	12.2337	0.399657	0.199829	−	0.979831i	$-0.435961\pi$
$937$	12.2337	0.399657	0.199829	+	0.979831i	$0.435961\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.0951	−1.50266	−0.751329	−	0.659928i	$-0.770587\pi$
$941$	−46.0951	−1.50266	−0.751329	+	0.659928i	$0.770587\pi$
$942$	0	0
$943$	2.20979	0.0719605
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.0986	1.10805	0.554027	−	0.832499i	$-0.313090\pi$
$947$	34.0986	1.10805	0.554027	+	0.832499i	$0.313090\pi$
$948$	0	0
$949$	20.2337	0.656813
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31.6060	−1.02382	−0.511909	−	0.859040i	$-0.671061\pi$
$953$	−31.6060	−1.02382	−0.511909	+	0.859040i	$0.671061\pi$
$954$	0	0
$955$	40.5988	1.31375
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−27.5983	−0.891196
$960$	0	0
$961$	62.3505	2.01131
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.37228	0.0763664
$966$	0	0
$967$	−27.5983	−0.887501	−0.443751	−	0.896150i	$-0.646352\pi$
$967$	−27.5983	−0.887501	−0.443751	+	0.896150i	$0.646352\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49.1317	1.57671	0.788356	−	0.615220i	$-0.210933\pi$
$971$	49.1317	1.57671	0.788356	+	0.615220i	$0.210933\pi$
$972$	0	0
$973$	6.41578	0.205680
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.7663	0.344445	0.172222	−	0.985058i	$-0.444905\pi$
$977$	10.7663	0.344445	0.172222	+	0.985058i	$0.444905\pi$
$978$	0	0
$979$	−7.45202	−0.238168
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.5988	1.29490	0.647451	−	0.762107i	$-0.275835\pi$
$983$	40.5988	1.29490	0.647451	+	0.762107i	$0.275835\pi$
$984$	0	0
$985$	1.76631	0.0562794
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.1168	0.417091
$990$	0	0
$991$	−47.4864	−1.50845	−0.754227	−	0.656613i	$-0.771988\pi$
$991$	−47.4864	−1.50845	−0.754227	+	0.656613i	$0.771988\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	49.1317	1.55758
$996$	0	0
$997$	41.3505	1.30958	0.654792	−	0.755809i	$-0.272756\pi$
$997$	41.3505	1.30958	0.654792	+	0.755809i	$0.272756\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.a.u.1.4		4
3.2	odd	2		2592.2.a.x.1.2		4
4.3	odd	2	inner	2592.2.a.u.1.3		4
8.3	odd	2		5184.2.a.cf.1.1		4
8.5	even	2		5184.2.a.cf.1.2		4
9.2	odd	6		864.2.i.f.577.3		8
9.4	even	3		288.2.i.f.97.1	✓	8
9.5	odd	6		864.2.i.f.289.3		8
9.7	even	3		288.2.i.f.193.1	yes	8
12.11	even	2		2592.2.a.x.1.1		4
24.5	odd	2		5184.2.a.cc.1.4		4
24.11	even	2		5184.2.a.cc.1.3		4
36.7	odd	6		288.2.i.f.193.4	yes	8
36.11	even	6		864.2.i.f.577.4		8
36.23	even	6		864.2.i.f.289.4		8
36.31	odd	6		288.2.i.f.97.4	yes	8
72.5	odd	6		1728.2.i.n.1153.1		8
72.11	even	6		1728.2.i.n.577.2		8
72.13	even	6		576.2.i.n.385.4		8
72.29	odd	6		1728.2.i.n.577.1		8
72.43	odd	6		576.2.i.n.193.1		8
72.59	even	6		1728.2.i.n.1153.2		8
72.61	even	6		576.2.i.n.193.4		8
72.67	odd	6		576.2.i.n.385.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.i.f.97.1	✓	8	9.4	even	3
288.2.i.f.97.4	yes	8	36.31	odd	6
288.2.i.f.193.1	yes	8	9.7	even	3
288.2.i.f.193.4	yes	8	36.7	odd	6
576.2.i.n.193.1		8	72.43	odd	6
576.2.i.n.193.4		8	72.61	even	6
576.2.i.n.385.1		8	72.67	odd	6
576.2.i.n.385.4		8	72.13	even	6
864.2.i.f.289.3		8	9.5	odd	6
864.2.i.f.289.4		8	36.23	even	6
864.2.i.f.577.3		8	9.2	odd	6
864.2.i.f.577.4		8	36.11	even	6
1728.2.i.n.577.1		8	72.29	odd	6
1728.2.i.n.577.2		8	72.11	even	6
1728.2.i.n.1153.1		8	72.5	odd	6
1728.2.i.n.1153.2		8	72.59	even	6
2592.2.a.u.1.3		4	4.3	odd	2	inner
2592.2.a.u.1.4		4	1.1	even	1	trivial
2592.2.a.x.1.1		4	12.11	even	2
2592.2.a.x.1.2		4	3.2	odd	2
5184.2.a.cc.1.3		4	24.11	even	2
5184.2.a.cc.1.4		4	24.5	odd	2
5184.2.a.cf.1.1		4	8.3	odd	2
5184.2.a.cf.1.2		4	8.5	even	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37228 q^{5} +2.20979 q^{7} -5.93580 q^{11} +4.37228 q^{13} +3.37228 q^{17} -3.72601 q^{19} +2.20979 q^{23} +0.627719 q^{25} +0.372281 q^{29} +9.66181 q^{31} +5.24224 q^{35} +4.00000 q^{37} +1.00000 q^{41} +5.93580 q^{43} +2.20979 q^{47} -2.11684 q^{49} -4.00000 q^{53} -14.0814 q^{55} +10.3554 q^{59} +15.1168 q^{61} +10.3723 q^{65} -10.3554 q^{67} +4.41957 q^{71} +4.62772 q^{73} -13.1168 q^{77} -9.66181 q^{79} -14.0814 q^{83} +8.00000 q^{85} +1.25544 q^{89} +9.66181 q^{91} -8.83915 q^{95} +9.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 6 q^{13} + 2 q^{17} + 14 q^{25} - 10 q^{29} + 16 q^{37} + 4 q^{41} + 26 q^{49} - 16 q^{53} + 26 q^{61} + 30 q^{65} + 30 q^{73} - 18 q^{77} + 32 q^{85} + 28 q^{89} + 36 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 6 * q^13 + 2 * q^17 + 14 * q^25 - 10 * q^29 + 16 * q^37 + 4 * q^41 + 26 * q^49 - 16 * q^53 + 26 * q^61 + 30 * q^65 + 30 * q^73 - 18 * q^77 + 32 * q^85 + 28 * q^89 + 36 * q^97

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