LMFDB - Embedded newform 472.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [472,2,Mod(1,472)] 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("472.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 472 = 2^{3} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-1,0,-1,0,1,0,-2,0,4] 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:	$3.76893897540$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +4.00000 q^{11} +2.00000 q^{13} +1.00000 q^{15} +2.00000 q^{17} +3.00000 q^{19} -1.00000 q^{21} +6.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} +5.00000 q^{29} +4.00000 q^{31} -4.00000 q^{33} -1.00000 q^{35} -6.00000 q^{37} -2.00000 q^{39} +3.00000 q^{41} +8.00000 q^{43} +2.00000 q^{45} -2.00000 q^{47} -6.00000 q^{49} -2.00000 q^{51} +11.0000 q^{53} -4.00000 q^{55} -3.00000 q^{57} +1.00000 q^{59} -2.00000 q^{63} -2.00000 q^{65} -8.00000 q^{67} -6.00000 q^{69} -8.00000 q^{71} -6.00000 q^{73} +4.00000 q^{75} +4.00000 q^{77} -1.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -2.00000 q^{85} -5.00000 q^{87} -16.0000 q^{89} +2.00000 q^{91} -4.00000 q^{93} -3.00000 q^{95} -10.0000 q^{97} -8.00000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−3.00000	−0.307794
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−8.00000	−0.804030
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	−5.00000	−0.415227
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	−11.0000	−0.872357
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−1.00000	−0.0751646
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	27.0000	1.94350	0.971751	−	0.236007i	$-0.0758390\pi$
$193$	27.0000	1.94350	0.971751	+	0.236007i	$0.0758390\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	5.00000	0.350931
$204$	0	0
$205$	−3.00000	−0.209529
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	1.00000	0.0649570
$238$	0	0
$239$	21.0000	1.35838	0.679189	−	0.733964i	$-0.262332\pi$
$239$	21.0000	1.35838	0.679189	+	0.733964i	$0.262332\pi$
$240$	0	0
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	−11.0000	−0.686161	−0.343081	−	0.939306i	$-0.611470\pi$
$257$	−11.0000	−0.686161	−0.343081	+	0.939306i	$0.611470\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−10.0000	−0.618984
$262$	0	0
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	−11.0000	−0.675725
$266$	0	0
$267$	16.0000	0.979184
$268$	0	0
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	0	0
$271$	−21.0000	−1.27566	−0.637830	−	0.770178i	$-0.720168\pi$
$271$	−21.0000	−1.27566	−0.637830	+	0.770178i	$0.720168\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	−16.0000	−0.964836
$276$	0	0
$277$	27.0000	1.62227	0.811136	−	0.584857i	$-0.198849\pi$
$277$	27.0000	1.62227	0.811136	+	0.584857i	$0.198849\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	29.0000	1.69420	0.847099	−	0.531435i	$-0.178347\pi$
$293$	29.0000	1.69420	0.847099	+	0.531435i	$0.178347\pi$
$294$	0	0
$295$	−1.00000	−0.0582223
$296$	0	0
$297$	20.0000	1.16052
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	0	0
$311$	5.00000	0.283524	0.141762	−	0.989901i	$-0.454723\pi$
$311$	5.00000	0.283524	0.141762	+	0.989901i	$0.454723\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	−8.00000	−0.443760
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	−8.00000	−0.434500
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	10.0000	0.533761
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	−2.00000	−0.105851
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	11.0000	0.571092
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	10.0000	0.515026
$378$	0	0
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−16.0000	−0.813326
$388$	0	0
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	10.0000	0.504433
$394$	0	0
$395$	1.00000	0.0503155
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	0	0
$413$	1.00000	0.0492068
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	−29.0000	−1.39365	−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000	−1.39365	−0.696826	+	0.717241i	$0.745405\pi$
$434$	0	0
$435$	5.00000	0.239732
$436$	0	0
$437$	18.0000	0.861057
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	10.0000	0.469841
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	10.0000	0.466760
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	32.0000	1.47136
$474$	0	0
$475$	−12.0000	−0.550598
$476$	0	0
$477$	−22.0000	−1.00731
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	−3.00000	−0.135943	−0.0679715	−	0.997687i	$-0.521653\pi$
$487$	−3.00000	−0.135943	−0.0679715	+	0.997687i	$0.521653\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	0	0
$493$	10.0000	0.450377
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	−21.0000	−0.940089	−0.470045	−	0.882643i	$-0.655762\pi$
$499$	−21.0000	−0.940089	−0.470045	+	0.882643i	$0.655762\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	0	0
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−44.0000	−1.95027	−0.975133	−	0.221621i	$-0.928865\pi$
$509$	−44.0000	−1.95027	−0.975133	+	0.221621i	$0.928865\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	15.0000	0.662266
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	−8.00000	−0.351840
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	25.0000	1.09317	0.546587	−	0.837402i	$-0.315927\pi$
$523$	25.0000	1.09317	0.546587	+	0.837402i	$0.315927\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−2.00000	−0.0867926
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	−3.00000	−0.129701
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	−24.0000	−1.03375
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	11.0000	0.472055
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15.0000	0.639021
$552$	0	0
$553$	−1.00000	−0.0425243
$554$	0	0
$555$	−6.00000	−0.254686
$556$	0	0
$557$	−33.0000	−1.39825	−0.699127	−	0.714997i	$-0.746428\pi$
$557$	−33.0000	−1.39825	−0.699127	+	0.714997i	$0.746428\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	−10.0000	−0.417756
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	37.0000	1.54033	0.770165	−	0.637845i	$-0.220174\pi$
$577$	37.0000	1.54033	0.770165	+	0.637845i	$0.220174\pi$
$578$	0	0
$579$	−27.0000	−1.12208
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	44.0000	1.82229
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	44.0000	1.81607	0.908037	−	0.418890i	$-0.137581\pi$
$587$	44.0000	1.81607	0.908037	+	0.418890i	$0.137581\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	37.0000	1.51941	0.759704	−	0.650269i	$-0.225344\pi$
$593$	37.0000	1.51941	0.759704	+	0.650269i	$0.225344\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	0	0
$597$	−17.0000	−0.695764
$598$	0	0
$599$	−25.0000	−1.02147	−0.510736	−	0.859738i	$-0.670627\pi$
$599$	−25.0000	−1.02147	−0.510736	+	0.859738i	$0.670627\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	16.0000	0.651570
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−7.00000	−0.284121	−0.142061	−	0.989858i	$-0.545373\pi$
$607$	−7.00000	−0.284121	−0.142061	+	0.989858i	$0.545373\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	3.00000	0.120972
$616$	0	0
$617$	1.00000	0.0402585	0.0201292	−	0.999797i	$-0.493592\pi$
$617$	1.00000	0.0402585	0.0201292	+	0.999797i	$0.493592\pi$
$618$	0	0
$619$	9.00000	0.361741	0.180870	−	0.983507i	$-0.442109\pi$
$619$	9.00000	0.361741	0.180870	+	0.983507i	$0.442109\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	0	0
$623$	−16.0000	−0.641026
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−7.00000	−0.277787
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	16.0000	0.632950
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	0	0
$643$	−49.0000	−1.93237	−0.966186	−	0.257847i	$-0.916987\pi$
$643$	−49.0000	−1.93237	−0.966186	+	0.257847i	$0.916987\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	9.00000	0.353827	0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000	0.353827	0.176913	+	0.984226i	$0.443389\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	0	0
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	30.0000	1.16160
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	0	0
$685$	9.00000	0.343872
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	22.0000	0.838133
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	0	0
$693$	−8.00000	−0.303895
$694$	0	0
$695$	16.0000	0.606915
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	0	0
$705$	−2.00000	−0.0753244
$706$	0	0
$707$	2.00000	0.0752177
$708$	0	0
$709$	−23.0000	−0.863783	−0.431892	−	0.901926i	$-0.642154\pi$
$709$	−23.0000	−0.863783	−0.431892	+	0.901926i	$0.642154\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	−21.0000	−0.784259
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	0	0
$723$	15.0000	0.557856
$724$	0	0
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	−32.0000	−1.17874
$738$	0	0
$739$	22.0000	0.809283	0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	3.00000	0.109618
$750$	0	0
$751$	46.0000	1.67856	0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000	1.67856	0.839282	+	0.543696i	$0.182976\pi$
$752$	0	0
$753$	15.0000	0.546630
$754$	0	0
$755$	10.0000	0.363937
$756$	0	0
$757$	49.0000	1.78094	0.890468	−	0.455047i	$-0.150377\pi$
$757$	49.0000	1.78094	0.890468	+	0.455047i	$0.150377\pi$
$758$	0	0
$759$	−24.0000	−0.871145
$760$	0	0
$761$	5.00000	0.181250	0.0906249	−	0.995885i	$-0.471114\pi$
$761$	5.00000	0.181250	0.0906249	+	0.995885i	$0.471114\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	2.00000	0.0722158
$768$	0	0
$769$	28.0000	1.00971	0.504853	−	0.863205i	$-0.331547\pi$
$769$	28.0000	1.00971	0.504853	+	0.863205i	$0.331547\pi$
$770$	0	0
$771$	11.0000	0.396155
$772$	0	0
$773$	−20.0000	−0.719350	−0.359675	−	0.933078i	$-0.617112\pi$
$773$	−20.0000	−0.719350	−0.359675	+	0.933078i	$0.617112\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	9.00000	0.322458
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	25.0000	0.893427
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	0	0
$789$	15.0000	0.534014
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	0	0
$794$	0	0
$795$	11.0000	0.390130
$796$	0	0
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	32.0000	1.13066
$802$	0	0
$803$	−24.0000	−0.846942
$804$	0	0
$805$	−6.00000	−0.211472
$806$	0	0
$807$	−8.00000	−0.281613
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	21.0000	0.736502
$814$	0	0
$815$	−16.0000	−0.560456
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−56.0000	−1.95441	−0.977207	−	0.212290i	$-0.931908\pi$
$821$	−56.0000	−1.95441	−0.977207	+	0.212290i	$0.931908\pi$
$822$	0	0
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	16.0000	0.557048
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	53.0000	1.84077	0.920383	−	0.391018i	$-0.127877\pi$
$829$	53.0000	1.84077	0.920383	+	0.391018i	$0.127877\pi$
$830$	0	0
$831$	−27.0000	−0.936620
$832$	0	0
$833$	−12.0000	−0.415775
$834$	0	0
$835$	−21.0000	−0.726735
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−15.0000	−0.516627
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−36.0000	−1.23406
$852$	0	0
$853$	−11.0000	−0.376633	−0.188316	−	0.982108i	$-0.560303\pi$
$853$	−11.0000	−0.376633	−0.188316	+	0.982108i	$0.560303\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−38.0000	−1.29654	−0.648272	−	0.761409i	$-0.724508\pi$
$859$	−38.0000	−1.29654	−0.648272	+	0.761409i	$0.724508\pi$
$860$	0	0
$861$	−3.00000	−0.102240
$862$	0	0
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	20.0000	0.676897
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	−43.0000	−1.45201	−0.726003	−	0.687691i	$-0.758624\pi$
$877$	−43.0000	−1.45201	−0.726003	+	0.687691i	$0.758624\pi$
$878$	0	0
$879$	−29.0000	−0.978146
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	0	0
$885$	1.00000	0.0336146
$886$	0	0
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−6.00000	−0.200782
$894$	0	0
$895$	16.0000	0.534821
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	20.0000	0.667037
$900$	0	0
$901$	22.0000	0.732926
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	11.0000	0.365652
$906$	0	0
$907$	−33.0000	−1.09575	−0.547874	−	0.836561i	$-0.684562\pi$
$907$	−33.0000	−1.09575	−0.547874	+	0.836561i	$0.684562\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	45.0000	1.49092	0.745458	−	0.666552i	$-0.232231\pi$
$911$	45.0000	1.49092	0.745458	+	0.666552i	$0.232231\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.0000	−0.330229
$918$	0	0
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	−16.0000	−0.526646
$924$	0	0
$925$	24.0000	0.789115
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	−44.0000	−1.44359	−0.721797	−	0.692105i	$-0.756683\pi$
$929$	−44.0000	−1.44359	−0.721797	+	0.692105i	$0.756683\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	0	0
$933$	−5.00000	−0.163693
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	28.0000	0.913745
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	18.0000	0.586161
$944$	0	0
$945$	−5.00000	−0.162650
$946$	0	0
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	14.0000	0.453981
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	−10.0000	−0.323592
$956$	0	0
$957$	−20.0000	−0.646508
$958$	0	0
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	−27.0000	−0.869161
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−55.0000	−1.76503	−0.882517	−	0.470281i	$-0.844153\pi$
$971$	−55.0000	−1.76503	−0.882517	+	0.470281i	$0.844153\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	8.00000	0.256205
$976$	0	0
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	−64.0000	−2.04545
$980$	0	0
$981$	−20.0000	−0.638551
$982$	0	0
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	2.00000	0.0636607
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	10.0000	0.317660	0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000	0.317660	0.158830	+	0.987306i	$0.449228\pi$
$992$	0	0
$993$	−7.00000	−0.222138
$994$	0	0
$995$	−17.0000	−0.538936
$996$	0	0
$997$	−11.0000	−0.348373	−0.174187	−	0.984713i	$-0.555730\pi$
$997$	−11.0000	−0.348373	−0.174187	+	0.984713i	$0.555730\pi$
$998$	0	0
$999$	−30.0000	−0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	472.2.a.c.1.1	✓	1
3.2	odd	2		4248.2.a.d.1.1		1
4.3	odd	2		944.2.a.g.1.1		1
8.3	odd	2		3776.2.a.j.1.1		1
8.5	even	2		3776.2.a.q.1.1		1
12.11	even	2		8496.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
472.2.a.c.1.1	✓	1	1.1	even	1	trivial
944.2.a.g.1.1		1	4.3	odd	2
3776.2.a.j.1.1		1	8.3	odd	2
3776.2.a.q.1.1		1	8.5	even	2
4248.2.a.d.1.1		1	3.2	odd	2
8496.2.a.p.1.1		1	12.11	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:	\( N \)	\(=\)	\( 472 = 2^{3} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	472.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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