LMFDB - Embedded newform 8016.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -4.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -4.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +6.00000 q^{13} -4.00000 q^{15} -4.00000 q^{19} -4.00000 q^{21} +8.00000 q^{23} +11.0000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -4.00000 q^{33} +16.0000 q^{35} -6.00000 q^{37} +6.00000 q^{39} +6.00000 q^{43} -4.00000 q^{45} -8.00000 q^{47} +9.00000 q^{49} +12.0000 q^{53} +16.0000 q^{55} -4.00000 q^{57} -12.0000 q^{59} +2.00000 q^{61} -4.00000 q^{63} -24.0000 q^{65} +2.00000 q^{67} +8.00000 q^{69} +12.0000 q^{71} -2.00000 q^{73} +11.0000 q^{75} +16.0000 q^{77} +2.00000 q^{79} +1.00000 q^{81} -8.00000 q^{83} +6.00000 q^{87} -6.00000 q^{89} -24.0000 q^{91} +16.0000 q^{95} +10.0000 q^{97} -4.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$4$	0	0
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	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
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	16.0000	2.70449
$36$	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
$39$	6.00000	0.960769
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	−4.00000	−0.596285
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	16.0000	2.15744
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	−24.0000	−2.97683
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	11.0000	1.27017
$76$	0	0
$77$	16.0000	1.82337
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−24.0000	−2.51588
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.0000	1.64157
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	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
$105$	16.0000	1.56144
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	−32.0000	−2.98402
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−24.0000	−2.14663
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	−24.0000	−1.99309
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	0	0
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	−32.0000	−2.52195
$162$	0	0
$163$	18.0000	1.40987	0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000	1.40987	0.704934	+	0.709273i	$0.250976\pi$
$164$	0	0
$165$	16.0000	1.24560
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−44.0000	−3.32609
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	24.0000	1.76452
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−24.0000	−1.71868
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	−24.0000	−1.68447
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	−24.0000	−1.63679
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	32.0000	2.08745
$236$	0	0
$237$	2.00000	0.129914
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−36.0000	−2.29996
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	−32.0000	−2.01182
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	0	0
$259$	24.0000	1.49129
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−48.0000	−2.94862
$266$	0	0
$267$	−6.00000	−0.367194
$268$	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
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	0	0
$273$	−24.0000	−1.45255
$274$	0	0
$275$	−44.0000	−2.65330
$276$	0	0
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	16.0000	0.947758
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	48.0000	2.79467
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	48.0000	2.77591
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	16.0000	0.901498
$316$	0	0
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	66.0000	3.66102
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	−32.0000	−1.72282
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	−48.0000	−2.54758
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	8.00000	0.418739
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−48.0000	−2.49204
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	0	0
$377$	36.0000	1.85409
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	−64.0000	−3.26174
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−4.00000	−0.198762
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	48.0000	2.36193
$414$	0	0
$415$	32.0000	1.57082
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	−24.0000	−1.15873
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	−24.0000	−1.15071
$436$	0	0
$437$	−32.0000	−1.53077
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	−8.00000	−0.378387
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	22.0000	1.03365
$454$	0	0
$455$	96.0000	4.50055
$456$	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
$459$	0	0
$460$	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
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	−44.0000	−2.01886
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	0	0
$483$	−32.0000	−1.45605
$484$	0	0
$485$	−40.0000	−1.81631
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	0	0
$489$	18.0000	0.813988
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	16.0000	0.719147
$496$	0	0
$497$	−48.0000	−2.15309
$498$	0	0
$499$	18.0000	0.805791	0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000	0.805791	0.402895	+	0.915246i	$0.368004\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	16.0000	0.711991
$506$	0	0
$507$	23.0000	1.02147
$508$	0	0
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	24.0000	1.05757
$516$	0	0
$517$	32.0000	1.40736
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	28.0000	1.22670	0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000	1.22670	0.613351	+	0.789810i	$0.289821\pi$
$522$	0	0
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	0	0
$525$	−44.0000	−1.92032
$526$	0	0
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	48.0000	2.07522
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−36.0000	−1.55063
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	24.0000	1.01874
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	36.0000	1.52264
$560$	0	0
$561$	0	0
$562$	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
$565$	48.0000	2.01938
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	40.0000	1.67689	0.838444	−	0.544988i	$-0.183466\pi$
$569$	40.0000	1.67689	0.838444	+	0.544988i	$0.183466\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	88.0000	3.66985
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	32.0000	1.32758
$582$	0	0
$583$	−48.0000	−1.98796
$584$	0	0
$585$	−24.0000	−0.992278
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−8.00000	−0.329076
$592$	0	0
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	−20.0000	−0.813116
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	−48.0000	−1.94187
$612$	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
$615$	0	0
$616$	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
$619$	38.0000	1.52735	0.763674	−	0.645601i	$-0.223393\pi$
$619$	38.0000	1.52735	0.763674	+	0.645601i	$0.223393\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	54.0000	2.13956
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	0	0
$643$	30.0000	1.18308	0.591542	−	0.806274i	$-0.298519\pi$
$643$	30.0000	1.18308	0.591542	+	0.806274i	$0.298519\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	0	0
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−64.0000	−2.48181
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	20.0000	0.773245
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	11.0000	0.423390
$676$	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
$679$	−40.0000	−1.53506
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	0	0
$687$	26.0000	0.991962
$688$	0	0
$689$	72.0000	2.74298
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	0	0
$693$	16.0000	0.607790
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	32.0000	1.20519
$706$	0	0
$707$	16.0000	0.601742
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	0	0
$714$	0	0
$715$	96.0000	3.59020
$716$	0	0
$717$	8.00000	0.298765
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	6.00000	0.223142
$724$	0	0
$725$	66.0000	2.45118
$726$	0	0
$727$	−22.0000	−0.815935	−0.407967	−	0.912996i	$-0.633762\pi$
$727$	−22.0000	−0.815935	−0.407967	+	0.912996i	$0.633762\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	0	0
$735$	−36.0000	−1.32788
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	32.0000	1.17239
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	48.0000	1.75388
$750$	0	0
$751$	30.0000	1.09472	0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000	1.09472	0.547358	+	0.836899i	$0.315634\pi$
$752$	0	0
$753$	−28.0000	−1.02038
$754$	0	0
$755$	−88.0000	−3.20265
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−72.0000	−2.59977
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	−16.0000	−0.576226
$772$	0	0
$773$	−32.0000	−1.15096	−0.575480	−	0.817816i	$-0.695185\pi$
$773$	−32.0000	−1.15096	−0.575480	+	0.817816i	$0.695185\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	72.0000	2.56979
$786$	0	0
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	−48.0000	−1.70238
$796$	0	0
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	8.00000	0.282314
$804$	0	0
$805$	128.000	4.51141
$806$	0	0
$807$	8.00000	0.281613
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	−30.0000	−1.05215
$814$	0	0
$815$	−72.0000	−2.52205
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	−24.0000	−0.838628
$820$	0	0
$821$	32.0000	1.11681	0.558404	−	0.829569i	$-0.311414\pi$
$821$	32.0000	1.11681	0.558404	+	0.829569i	$0.311414\pi$
$822$	0	0
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	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
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	30.0000	1.04069
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.00000	0.138426
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−2.00000	−0.0688837
$844$	0	0
$845$	−92.0000	−3.16490
$846$	0	0
$847$	−20.0000	−0.687208
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	72.0000	2.44807
$866$	0	0
$867$	−17.0000	−0.577350
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	96.0000	3.24539
$876$	0	0
$877$	−58.0000	−1.95852	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−58.0000	−1.95852	−0.979260	+	0.202606i	$0.935059\pi$
$878$	0	0
$879$	−10.0000	−0.337292
$880$	0	0
$881$	−20.0000	−0.673817	−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000	−0.673817	−0.336909	+	0.941537i	$0.609381\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	48.0000	1.61350
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	48.0000	1.60267
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	56.0000	1.86150
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	0	0
$909$	−4.00000	−0.132672
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	72.0000	2.36991
$924$	0	0
$925$	−66.0000	−2.17007
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	16.0000	0.520480
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	14.0000	0.453981
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−24.0000	−0.775810
$958$	0	0
$959$	8.00000	0.258333
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	−8.00000	−0.257529
$966$	0	0
$967$	−12.0000	−0.385894	−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000	−0.385894	−0.192947	+	0.981209i	$0.561805\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	0	0
$975$	66.0000	2.11369
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	−28.0000	−0.893061	−0.446531	−	0.894768i	$-0.647341\pi$
$983$	−28.0000	−0.893061	−0.446531	+	0.894768i	$0.647341\pi$
$984$	0	0
$985$	32.0000	1.01960
$986$	0	0
$987$	32.0000	1.01857
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	18.0000	0.571789	0.285894	−	0.958261i	$-0.407709\pi$
$991$	18.0000	0.571789	0.285894	+	0.958261i	$0.407709\pi$
$992$	0	0
$993$	−26.0000	−0.825085
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.g.1.1		1
4.3	odd	2		501.2.a.a.1.1	✓	1
12.11	even	2		1503.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.a.1.1	✓	1	4.3	odd	2
1503.2.a.a.1.1		1	12.11	even	2
8016.2.a.g.1.1		1	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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