LMFDB - Embedded newform 8464.2.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,4,0,2,0,-2,0,-4,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +4.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} -4.00000 q^{11} -5.00000 q^{13} +4.00000 q^{15} +2.00000 q^{17} +6.00000 q^{19} +2.00000 q^{21} +11.0000 q^{25} -5.00000 q^{27} +1.00000 q^{29} +9.00000 q^{31} -4.00000 q^{33} +8.00000 q^{35} +4.00000 q^{37} -5.00000 q^{39} +3.00000 q^{41} +8.00000 q^{43} -8.00000 q^{45} +5.00000 q^{47} -3.00000 q^{49} +2.00000 q^{51} -6.00000 q^{53} -16.0000 q^{55} +6.00000 q^{57} +4.00000 q^{59} +10.0000 q^{61} -4.00000 q^{63} -20.0000 q^{65} -4.00000 q^{67} +5.00000 q^{71} -15.0000 q^{73} +11.0000 q^{75} -8.00000 q^{77} -6.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +8.00000 q^{85} +1.00000 q^{87} +8.00000 q^{89} -10.0000 q^{91} +9.00000 q^{93} +24.0000 q^{95} -10.0000 q^{97} +8.00000 q^{99} +O(q^{100})$

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	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	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
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	8.00000	1.35225
$36$	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
$39$	−5.00000	−0.800641
$40$	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
$45$	−8.00000	−1.19257
$46$	0	0
$47$	5.00000	0.729325	0.364662	−	0.931140i	$-0.381184\pi$
$47$	5.00000	0.729325	0.364662	+	0.931140i	$0.381184\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−16.0000	−2.15744
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	−20.0000	−2.48069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	0	0
$75$	11.0000	1.27017
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	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
$85$	8.00000	0.867722
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	−10.0000	−1.04828
$92$	0	0
$93$	9.00000	0.933257
$94$	0	0
$95$	24.0000	2.46235
$96$	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
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.0000	0.924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	3.00000	0.270501
$124$	0	0
$125$	24.0000	2.14663
$126$	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
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	−20.0000	−1.72133
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	0	0
$143$	20.0000	1.67248
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	36.0000	2.89159
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−25.0000	−1.95815	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−25.0000	−1.95815	−0.979076	+	0.203497i	$0.934769\pi$
$164$	0	0
$165$	−16.0000	−1.24560
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−12.0000	−0.917663
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	22.0000	1.66304
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−1.00000	−0.0747435	−0.0373718	−	0.999301i	$-0.511899\pi$
$179$	−1.00000	−0.0747435	−0.0373718	+	0.999301i	$0.511899\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	16.0000	1.17634
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−10.0000	−0.727393
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	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
$195$	−20.0000	−1.43223
$196$	0	0
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−24.0000	−1.66011
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	5.00000	0.342594
$214$	0	0
$215$	32.0000	2.18238
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	−15.0000	−1.01361
$220$	0	0
$221$	−10.0000	−0.672673
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	−22.0000	−1.46667
$226$	0	0
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	20.0000	1.30466
$236$	0	0
$237$	−6.00000	−0.389742
$238$	0	0
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	0	0
$241$	16.0000	1.03065	0.515325	−	0.856995i	$-0.327671\pi$
$241$	16.0000	1.03065	0.515325	+	0.856995i	$0.327671\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	−12.0000	−0.766652
$246$	0	0
$247$	−30.0000	−1.90885
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	8.00000	0.500979
$256$	0	0
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	22.0000	1.35658	0.678289	−	0.734795i	$-0.262722\pi$
$263$	22.0000	1.35658	0.678289	+	0.734795i	$0.262722\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	0	0
$267$	8.00000	0.489592
$268$	0	0
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−10.0000	−0.605228
$274$	0	0
$275$	−44.0000	−2.65330
$276$	0	0
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	24.0000	1.42164
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−20.0000	−1.16841	−0.584206	−	0.811605i	$-0.698594\pi$
$293$	−20.0000	−1.16841	−0.584206	+	0.811605i	$0.698594\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	20.0000	1.16052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	40.0000	2.29039
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	0	0
$315$	−16.0000	−0.901498
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−55.0000	−3.05085
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.0000	0.551318
$330$	0	0
$331$	23.0000	1.26419	0.632097	−	0.774889i	$-0.282194\pi$
$331$	23.0000	1.26419	0.632097	+	0.774889i	$0.282194\pi$
$332$	0	0
$333$	−8.00000	−0.438397
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−36.0000	−1.94951
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	25.0000	1.33440
$352$	0	0
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	0	0
$355$	20.0000	1.06149
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−60.0000	−3.14054
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	8.00000	0.414224	0.207112	−	0.978317i	$-0.433593\pi$
$373$	8.00000	0.414224	0.207112	+	0.978317i	$0.433593\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	−5.00000	−0.257513
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	−32.0000	−1.63087
$386$	0	0
$387$	−16.0000	−0.813326
$388$	0	0
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	21.0000	1.05931
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	−45.0000	−2.24161
$404$	0	0
$405$	4.00000	0.198762
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−10.0000	−0.486217
$424$	0	0
$425$	22.0000	1.06716
$426$	0	0
$427$	20.0000	0.967868
$428$	0	0
$429$	20.0000	0.965609
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	5.00000	0.237557	0.118779	−	0.992921i	$-0.462102\pi$
$443$	5.00000	0.237557	0.118779	+	0.992921i	$0.462102\pi$
$444$	0	0
$445$	32.0000	1.51695
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	−11.0000	−0.516825
$454$	0	0
$455$	−40.0000	−1.87523
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	−10.0000	−0.466760
$460$	0	0
$461$	−7.00000	−0.326023	−0.163011	−	0.986624i	$-0.552121\pi$
$461$	−7.00000	−0.326023	−0.163011	+	0.986624i	$0.552121\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	36.0000	1.66946
$466$	0	0
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	12.0000	0.552931
$472$	0	0
$473$	−32.0000	−1.47136
$474$	0	0
$475$	66.0000	3.02829
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−40.0000	−1.81631
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	0	0
$489$	−25.0000	−1.13054
$490$	0	0
$491$	11.0000	0.496423	0.248212	−	0.968706i	$-0.420157\pi$
$491$	11.0000	0.496423	0.248212	+	0.968706i	$0.420157\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	32.0000	1.43829
$496$	0	0
$497$	10.0000	0.448561
$498$	0	0
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−22.0000	−0.980932	−0.490466	−	0.871460i	$-0.663173\pi$
$503$	−22.0000	−0.980932	−0.490466	+	0.871460i	$0.663173\pi$
$504$	0	0
$505$	−40.0000	−1.77998
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	0	0
$511$	−30.0000	−1.32712
$512$	0	0
$513$	−30.0000	−1.32453
$514$	0	0
$515$	40.0000	1.76261
$516$	0	0
$517$	−20.0000	−0.879599
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	0	0
$525$	22.0000	0.960159
$526$	0	0
$527$	18.0000	0.784092
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−15.0000	−0.649722
$534$	0	0
$535$	24.0000	1.03761
$536$	0	0
$537$	−1.00000	−0.0431532
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.00000	−0.384812	−0.192406	−	0.981315i	$-0.561629\pi$
$547$	−9.00000	−0.384812	−0.192406	+	0.981315i	$0.561629\pi$
$548$	0	0
$549$	−20.0000	−0.853579
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−12.0000	−0.510292
$554$	0	0
$555$	16.0000	0.679162
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	−24.0000	−1.00969
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	0	0
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−39.0000	−1.62359	−0.811796	−	0.583942i	$-0.801510\pi$
$577$	−39.0000	−1.62359	−0.811796	+	0.583942i	$0.801510\pi$
$578$	0	0
$579$	1.00000	0.0415586
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	40.0000	1.65380
$586$	0	0
$587$	13.0000	0.536567	0.268284	−	0.963340i	$-0.413544\pi$
$587$	13.0000	0.536567	0.268284	+	0.963340i	$0.413544\pi$
$588$	0	0
$589$	54.0000	2.22503
$590$	0	0
$591$	21.0000	0.863825
$592$	0	0
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	29.0000	1.18293	0.591467	−	0.806329i	$-0.298549\pi$
$601$	29.0000	1.18293	0.591467	+	0.806329i	$0.298549\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	20.0000	0.813116
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	−25.0000	−1.01139
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	−24.0000	−0.958468
$628$	0	0
$629$	8.00000	0.318981
$630$	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
$633$	16.0000	0.635943
$634$	0	0
$635$	28.0000	1.11115
$636$	0	0
$637$	15.0000	0.594322
$638$	0	0
$639$	−10.0000	−0.395594
$640$	0	0
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	32.0000	1.26000
$646$	0	0
$647$	−11.0000	−0.432455	−0.216227	−	0.976343i	$-0.569375\pi$
$647$	−11.0000	−0.432455	−0.216227	+	0.976343i	$0.569375\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	18.0000	0.705476
$652$	0	0
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	0	0
$655$	84.0000	3.28215
$656$	0	0
$657$	30.0000	1.17041
$658$	0	0
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	−10.0000	−0.388368
$664$	0	0
$665$	48.0000	1.86136
$666$	0	0
$667$	0	0
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−40.0000	−1.54418
$672$	0	0
$673$	43.0000	1.65753	0.828764	−	0.559598i	$-0.189045\pi$
$673$	43.0000	1.65753	0.828764	+	0.559598i	$0.189045\pi$
$674$	0	0
$675$	−55.0000	−2.11695
$676$	0	0
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−14.0000	−0.536481
$682$	0	0
$683$	31.0000	1.18618	0.593091	−	0.805135i	$-0.297907\pi$
$683$	31.0000	1.18618	0.593091	+	0.805135i	$0.297907\pi$
$684$	0	0
$685$	−16.0000	−0.611329
$686$	0	0
$687$	−28.0000	−1.06827
$688$	0	0
$689$	30.0000	1.14291
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	16.0000	0.607790
$694$	0	0
$695$	−20.0000	−0.758643
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	−9.00000	−0.340411
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	20.0000	0.753244
$706$	0	0
$707$	−20.0000	−0.752177
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	0	0
$714$	0	0
$715$	80.0000	2.99183
$716$	0	0
$717$	1.00000	0.0373457
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	16.0000	0.595046
$724$	0	0
$725$	11.0000	0.408530
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	7.00000	0.257499	0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000	0.257499	0.128750	+	0.991677i	$0.458904\pi$
$740$	0	0
$741$	−30.0000	−1.10208
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	72.0000	2.63788
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	0	0
$753$	30.0000	1.09326
$754$	0	0
$755$	−44.0000	−1.60132
$756$	0	0
$757$	−24.0000	−0.872295	−0.436147	−	0.899875i	$-0.643657\pi$
$757$	−24.0000	−0.872295	−0.436147	+	0.899875i	$0.643657\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−16.0000	−0.578481
$766$	0	0
$767$	−20.0000	−0.722158
$768$	0	0
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	0	0
$771$	−21.0000	−0.756297
$772$	0	0
$773$	48.0000	1.72644	0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000	1.72644	0.863220	+	0.504828i	$0.168444\pi$
$774$	0	0
$775$	99.0000	3.55618
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	48.0000	1.71319
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	22.0000	0.783221
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−50.0000	−1.77555
$794$	0	0
$795$	−24.0000	−0.851192
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	10.0000	0.353775
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	60.0000	2.11735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.0000	0.739235
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	−100.000	−3.50285
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	20.0000	0.698857
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	0	0
$825$	−44.0000	−1.53188
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	−7.00000	−0.242827
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−48.0000	−1.66111
$836$	0	0
$837$	−45.0000	−1.55543
$838$	0	0
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	48.0000	1.65125
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−48.0000	−1.64157
$856$	0	0
$857$	−1.00000	−0.0341593	−0.0170797	−	0.999854i	$-0.505437\pi$
$857$	−1.00000	−0.0341593	−0.0170797	+	0.999854i	$0.505437\pi$
$858$	0	0
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	20.0000	0.676897
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	−20.0000	−0.674583
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	−15.0000	−0.503651	−0.251825	−	0.967773i	$-0.581031\pi$
$887$	−15.0000	−0.503651	−0.251825	+	0.967773i	$0.581031\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	30.0000	1.00391
$894$	0	0
$895$	−4.00000	−0.133705
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.00000	0.300167
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	16.0000	0.532447
$904$	0	0
$905$	48.0000	1.59557
$906$	0	0
$907$	14.0000	0.464862	0.232431	−	0.972613i	$-0.425332\pi$
$907$	14.0000	0.464862	0.232431	+	0.972613i	$0.425332\pi$
$908$	0	0
$909$	20.0000	0.663358
$910$	0	0
$911$	−42.0000	−1.39152	−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000	−1.39152	−0.695761	+	0.718273i	$0.744933\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	0	0
$915$	40.0000	1.32236
$916$	0	0
$917$	42.0000	1.38696
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	−25.0000	−0.822885
$924$	0	0
$925$	44.0000	1.44671
$926$	0	0
$927$	−20.0000	−0.656886
$928$	0	0
$929$	−49.0000	−1.60764	−0.803819	−	0.594874i	$-0.797202\pi$
$929$	−49.0000	−1.60764	−0.803819	+	0.594874i	$0.797202\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	0	0
$933$	21.0000	0.687509
$934$	0	0
$935$	−32.0000	−1.04651
$936$	0	0
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−40.0000	−1.30120
$946$	0	0
$947$	−19.0000	−0.617417	−0.308709	−	0.951157i	$-0.599897\pi$
$947$	−19.0000	−0.617417	−0.308709	+	0.951157i	$0.599897\pi$
$948$	0	0
$949$	75.0000	2.43460
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	48.0000	1.55324
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	4.00000	0.128765
$966$	0	0
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	−10.0000	−0.320585
$974$	0	0
$975$	−55.0000	−1.76141
$976$	0	0
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.0000	−0.701691	−0.350846	−	0.936433i	$-0.614106\pi$
$983$	−22.0000	−0.701691	−0.350846	+	0.936433i	$0.614106\pi$
$984$	0	0
$985$	84.0000	2.67646
$986$	0	0
$987$	10.0000	0.318304
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	23.0000	0.729883
$994$	0	0
$995$	64.0000	2.02894
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	0	0
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.p.1.1		1
4.3	odd	2		4232.2.a.f.1.1		1
23.22	odd	2		368.2.a.e.1.1		1
69.68	even	2		3312.2.a.r.1.1		1
92.91	even	2		184.2.a.a.1.1	✓	1
115.114	odd	2		9200.2.a.o.1.1		1
184.45	odd	2		1472.2.a.e.1.1		1
184.91	even	2		1472.2.a.l.1.1		1
276.275	odd	2		1656.2.a.i.1.1		1
460.183	odd	4		4600.2.e.e.4049.1		2
460.367	odd	4		4600.2.e.e.4049.2		2
460.459	even	2		4600.2.a.i.1.1		1
644.643	odd	2		9016.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.a.a.1.1	✓	1	92.91	even	2
368.2.a.e.1.1		1	23.22	odd	2
1472.2.a.e.1.1		1	184.45	odd	2
1472.2.a.l.1.1		1	184.91	even	2
1656.2.a.i.1.1		1	276.275	odd	2
3312.2.a.r.1.1		1	69.68	even	2
4232.2.a.f.1.1		1	4.3	odd	2
4600.2.a.i.1.1		1	460.459	even	2
4600.2.e.e.4049.1		2	460.183	odd	4
4600.2.e.e.4049.2		2	460.367	odd	4
8464.2.a.p.1.1		1	1.1	even	1	trivial
9016.2.a.k.1.1		1	644.643	odd	2
9200.2.a.o.1.1		1	115.114	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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

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L-functions

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